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Fully spin-polarized open and closed nodal lines in β -borophene by
magnetic proximity effect
Article · September 2019
DOI: 10.1103/PhysRevB.100.115423
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PHYSICAL REVIEW B 100, 115423 (2019)
Fully spin-polarized open and closed nodal lines in β-borophene by magnetic proximity effect
Xi Zuo,1 A. C. Dias ,2 Fujun Liu ,2 Li Han,1 Heming Li,1 Quan Gao,1 Xinxin Jiang,1 Dongmei Li,1 Bin Cui ,1,*
Desheng Liu,1,3,† and Fanyao Qu 2
1
School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China
2
Instituto de Física, Universidade de Brasília, Brasília-DF 70919-970, Brazil
3
Department of Physics, Jining University, Qufu 273155, China
(Received 10 July 2019; revised manuscript received 29 August 2019; published 13 September 2019)
Recently, the experimental realization of Dirac fermions in β12 - and χ3 -boron sheets has attracted tremendous
attention and simulated a broad exploration for the novel topologically nontrivial states in two-dimensional
materials. Herein, by combining first-principles and tight-binding calculations, we discover a coexistence of
open nodal arc and closed nodal loop produced by three-band touching in the recently synthesized β-boron
sheet [Q. Zhong et al., Phys. Rev. Mater. 1, 021001 (2017)]. The symmetry analysis reveals that the nodal lines
are protected by both time-reversal symmetry (TRS) and mirror-reflection symmetry (MRS). Intriguingly, when
TRS is broken, e.g., via introducing magnetic proximity effect, a topological phase transition occurs, namely,
the original spin-degenerate nodal line (loop) decays into fully spin-polarized nodal line (loop). Furthermore,
when Rashba spin-orbit coupling (SOC) is turned on to break the MRS, spin-up and -down states are shifted
toward opposite momentum directions, leading to the emergence of two new nodal loops around the point. In
addition, inclusion of both Rashba SOC and exchange field leads to band-gap opening of the Dirac points and
nodal lines. Our findings not only reveal a form of nodal line in the β-boron sheet, but also offer an alternative
approach to realize spin-polarized nodal line semimetals with promising applications in spintronic devices.
DOI: 10.1103/PhysRevB.100.115423
I. INTRODUCTION
Topological semimetals (TSMs) [1,2], following the rising of topological insulators [3,4], have become one of the
most attractive areas in condensed matter physics. Different
from normal metals, the Fermi surface of TSMs consists
of nontrivial band-crossing points, called nodal points (NPs)
[5] due to the touching of conduction and valence bands in
momentum space (MS). According to the degeneracy and
distribution of the NPs, TSMs can be generally classified
into Dirac (DSMs) [6,7], Weyl (WSMs) [8,9], and nodal-line
semimetals (NLSMs) [5,10–12]. For DSMs and WSMs, the
NPs are discretely distributed in MS with fourfold and twofold
degeneracy, respectively. For NLSMs, the NPs form onedimensional nodal line in MS and have flexible topological
configurations in MS such as nodal ring [13], nodal chain
[14], and nodal nets [15]. Among these TSMs, NLSMs have
recently attracted a great deal of attention due to their distinct
drumheadlike surface flat bands [16,17] and compatibility
with other nontrivial fermions [18,19]. These unique properties endow NLSMs many exotic properties such as hightemperature surface superconductivity [20], nondispersive
Landau energy level [21], and specific long-range Coulomb
interactions [22], giving rise to its potential applications in
future low-dissipation electronic and spintronic devices.
Recently, the experimentally successful observation of the
Dirac fermion states in β12 - [23] and γ3 - [24] boron sheets
*
†
cuibin@sdu.edu.cn
liuds@sdu.edu.cn
2469-9950/2019/100(11)/115423(8)
has sparked intensive interest to explore the two-dimensional
(2D) topological semimetals (TSMs) [25–29]. Many exotic
phenomena such as the massless Dirac fermions with high
Fermi velocity, anisotropy Dirac cone have been predicted
in 8-Pmmn boron bilayer [30], honeycomb boron allotropes
[31], and Pmmn boron layer group [32]. Intriguingly, it has
also been revealed that the nodal-line fermions may exist in
systems such as 6B: P 6mmm boron bilayer [33], P 6-boron
bilayer [34], and striped boron sheet (hr-SB) [35]. However,
the complex layer structures or a mixture of different planar
phases severely hinder the experiment synthesis and observation of the nodal-line fermions in these proposed boron sheets,
which arouses tremendous efforts in exploring and identifying
TSMs, especially the NLSMs in monolayer 2D materials.
In this paper, we investigate the nontrivial topological
states in the β-borophene [36] synthesized recently. As shown
in Fig. 1, the unit cell of β-boron sheet contains seven
boron atoms and three inequivalent sites with 4-, 5-, and 6centered B-B bonds. Low-energy band structure possesses an
unconventional three-band-touching phenomenon on the -Y
high-symmetry line, resulting in a coexistence of closed nodal
loop and open nodal line in MS (Fig. 2). In addition, there are
three Dirac cones nearby the Fermi level with an ultrahigh
Fermi velocity (1.33 × 106 m/s) (Fig. 3). The nature of the
nodal lines is explored by performing symmetry analysis and
surface states calculation (Fig. 4). We find that both open
nodal line and nodal loop semimetal states are protected by
time-reversal symmetry (TRS) and mirror-reflection symmetry (MRS). Furthermore, we have also studied the possibility
of topological phase transition by breaking symmetry in this
system. Here, MRS is broken via application of a vertical
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©2019 American Physical Society
XI ZUO et al.
PHYSICAL REVIEW B 100, 115423 (2019)
electric field to the borophene, which induces Rashba spinorbit coupling (SOC). While TRS breaking is realized through
magnetic proximity effect [37] produced by ferromagnetic
substrate on which the boron sheet is deposited (Fig. 5).
Unlike conventional symmetry breaking approaches by virtue
of transition metal atoms doping [38,39] or atomic bulking [25–27], the way used here is an external manipulation
which does not require growth of doped samples or making
atomic bulking. Since the penetration depth of the magnetic
proximity effect is governed by short-range exchange interactions, it will become more pronounced in atomically thin
2D heterostructures [40,41]. In addition, from the perspective
of nanodevice application, the defects introduced by magnetic doping will reduce the mobility of devices [39]. We
find that magnetic exchange field endows nonmagnetic βborophene full spin-polarized nodal lines and loops, while the
Rashba SOC mixes spin-up and -down states around the bandcrossing points and shift them toward opposite momentum
directions, leading to the emergence of two new nodal loops
around the point (Fig. 6). In addition, the Dirac points and
nodal lines open gaps when both the Rashba SOC and spin
exchange field are present. Therefore, our results provide an
alternative scheme to engineer TSMs and shed light on the
application of NLSMs in spintronics.
II. COMPUTATIONAL DETAILS
Our calculations are performed using the projector augmented wave (PAW) method [42] implemented in the Vienna
ab initio simulation package (VASP) code [43,44]. The PerdewBurke-Ernzerhof (PBE) form of the generalized gradient approximation (GGA) exchange-correlation functional [45] is
adopted. The cutoff energy is set to 600 eV after the convergence tests. A -centered Monkhorst-Pack [46] k-point grid
of 25 × 11 × 1 for borophene unit cell is chosen for relaxations and the grid of 41 × 18 × 1 for property calculations.
In our calculations, the total energy is converged to less than
10−6 eV. The maximum force is less than 0.01 eV/Å during
optimization. A vacuum space between neighboring supercells is set to be more than 30 eV/Å to avoid spurious
interactions. The Fermi velocity near these Dirac cones is
calculated with the expression vF = E /h̄k, where the E /k is
the slope of the linear valence band (VB) or conduction band
(CB) and the h̄ is the reduced Planck’s constant.
For tight-binding calculations, we use the software WANNIER90 [47] for the Wannier interpolation, using a set of 21
Wannier orbitals (3 for each of the 7 atoms in the unit cell)
and the projections px , py , pz for each atom. To obtain proper
tight-binding parameters, a Monkhorst-Pack k-point grid of
10 × 10 × 1 has been adopted and 121 neighbor unit cells
have been used to make Wannier interpolation. After obtaining the tight-binding Hamiltonian, the Bloch spectrum with
edge contribution is further calculated by iterative Green’s
function method as implemented in the WANNIERTOOLS software [48].
FIG. 1. (a) The top and side views of the β-boron sheet. (b) The
2D Brillouin zone of β-borophene, high-symmetry points, and its
projected 1D Brillouin zone.
single rows of empty hexagons, which is similar to that of
β12 - and χ3 -boron sheets [36]. The symmetry of the β-boron
sheet is orthorhombic, which shares the same space group
Pmmm (No. 47) with the β12 phase. In Table I, we also
give a detailed structure parameter of the β-boron sheet and
the experimentally synthesized boron sheets for comparison.
The optimized lattice constants are in good accordance with
previously reported values [36,49,50], verifying the accuracy
of our GGA-PBE calculations. Compared with β12 - and χ3 boron sheets, the lattice constant of β-boron sheet is basically
unchanged in the x direction, but increases obviously in the
y direction, corresponding to the added B atoms in the y
direction of the filled boron chains. The bond length of βboron sheet ranges from 1.63 to 1.79 Å, which is very close to
those of β12 -, δ6 -, and χ3 -boron sheets.
The orbital projected band structure of the β-boron
sheet along high-symmetry line (-X -V --Y -V ) is shown in
Fig. 2(a). The results show that the bands near the Fermi
level are mainly contributed by px , py , and pz orbitals. Based
on the basis of the px , py , and pz orbitals, we also calculate
the band structure of the β-boron sheet by constructing a
tight-binding (TB) Hamiltonian from the maximally localized
Wannier functions (MLWFs). The obtained band structure
well fits with DFT results, verifying the accuracy of our
orbital component analysis for the β-boron sheet [see dashed
lines in Fig. 2(a)]. Intriguingly, three bands derived from px ,
py , and pz orbitals overlap at the point N1 near the Fermi
level with a large linear energy range along -Y direction
(>1 eV). A similar three-band touching has also been recently
TABLE I. Optimized lattice constants (a, b, γ ), bond length,
space group of the β-, δ6 -, β12 -, and χ3 -boron sheets. The α and β
angles of the unit cell are fixed at 90.0◦ .
Structure
a (Å)
b (Å)
γ◦
Bond length
Space group
βa
δ6 b
β12 c
χ3 d
2.94
1.61
2.93
2.91
6.73
2.87
5.07
4.45
90.00
90.00
90.00
70.90
1.63–1.79
1.61–1.88
1.65–1.75
1.62–1.72
Pmmm
Pmmn
Pmmm
Cmmm
a
III. RESULTS AND DISCUSSION
Reference [36].
Reference [49].
c
Reference [50].
d
Reference [50].
b
As shown in Fig. 1(a), the β-boron sheet has a planar
structure composed with filled boron chains separated by
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PHYSICAL REVIEW B 100, 115423 (2019)
FIG. 2. (a) Orbital projected band structure of β-boron sheet.
The orbital contributions and their weight are plotted corresponding
to different colors and spot size. The blue circles (D1-3) and squares
(N1-2) label different touching types of bands corresponding to Dirac
and nodal-line crossing points. The point group of all high-symmetry
points is D2h , and the irreducible representations of all symmetry
points are also given corresponding to the character table for D2h .
(b) 3D band structure on the kx -ky plane around N1. The red and
blue lines indicate the open nodal arcs. (c) The 2D contour of the
energy gap difference between the px and pz bands shown in (a).
The white line and lens-shaped white regions enclosed in the green
circles indicate the position of the nodal line and Dirac cones in the
2D Brillouin zone, respectively.
revealed by first-principles calculations [51] and tight-binding
methods [52] in β12 -borophene. At point, the py band
lies above the px and pz bands. However, at Y point, the pz
band lies above the px and py bands. The inversion of the
band ordering is usually accompanied with nontrivial states
protected by crystallographic symmetry. In order to have a
direct visualization of the three-band touching, we perform
a scan around the crossing point N1 along the -Y direction
as shown in Fig. 2(b). One can clearly observe that linearly
dispersed pz band crosses over the px and py bands along
the -Y direction, thus forming two open nodal arcs with a
tip to tip, which is obviously different from the closed nodal
lines predicted in recent investigations of boron polymorphs
[33,34,51,52]. Apart from the three-band touching along the
-Y direction, we also find that px and pz bands cross at N2
point. A careful examination of the energy difference between
the px and pz bands in Fig. 2(c) reveals that they actually form
a spindlelike nodal loop (see the white color line indicating
the zero-gap position) around the point. It is noteworthy
to mention that one nodal arc in the -Y direction is part of
the nodal loop formed by the crossing of px and pz bands.
Therefore, there is a coexistence of closed nodal loop and
open nodal arc in the β-boron sheet.
To gain a deeper insight into the origins of the open nodal
lines, we perform a symmetry analysis of the three bands
along -Y and -X directions. In the reciprocal space of the
Pmmm (No. 47), the high-symmetry points of , X , Y , and V
have D2h point group, which possesses three twofold rotation
axes (C2x , C2y , and C2z ) and three mirror planes (Mx , My ,
and Mz ). All the k points along the -X and -Y directions
share the point group of C2v with a common mirror-reflection
symmetry operator of Mz . With the help of group analysis,
we find that the irreducible representations (IRs) for the px ,
py , and pz bands are A1 , B1 , and B2 along the -Y direction.
Along the -X direction, the IRs for the px and pz bands are
A1 and B2 . Since the IRs of A1 , B1 , and B2 are even, even, and
odd under mirror-reflection symmetry, the mirror parities of px
and py bands are opposite to that of pz band. The opposite Mz
mirror symmetry indicates that pz band does not couple with
px and py bands, thus forming two nodal-line fermions around
the point, which is in accordance with the observations in
Fig. 2(b).
In Fig. 2(c), apart from the Dirac nodal lines, there are
several lens-shaped white regions enclosed in the green circles
locating at the terminal of -X and Y -V directions with mirror
counterparts at the corners or edges. This is consistent with the
Dirac-cone-like band-crossing points D1, D2, and D3 labeled
by blue circles at -X and Y -V directions [see Fig. 2(a)].
Similar to graphene, these bands are all contributed by pz
orbitals. Since the Dirac points act as sources and drains of the
Berry curvature flux, the Berry phase obtained by integrating
the Berry curvature over a closed surface around the Dirac
point must be π or −π [53,54] in two-dimensional MS. The
Berry phase of the nth state n(k) over a closed curve C (or
arbitrary surface S enclosed by the path C) in reciprocal space
k is defined as [48,55]
φn = dk · An (k)
C
= dS · ∇ k × An (k)
S
(1)
= dS · n (k),
S
∂
An (k) = in(k)| ∂k
|n(k)
where
is called the Berry vector
potential/Berry connection, and n (k) = ∇ k × An (k) represents the Berry curvature. To verify the nontrivial nature
of these crossing points, we calculate the Berry phases by
integrating the Berry vector potential around these crossing
points on a anticlockwise closed k path, which are exactly
quantized in units of π (π , −π , and π ) for D1, D2, and
D3, thus confirming they are Dirac points. Then, we use
group theory to demonstrate whether these Dirac points are
symmetry protected. At the X point, the IRs of two bands of
D1 (D2) are B2g (Au ) and B3g (B1u ) with opposite eigenvalues
under symmetry operators My and Mx . Thus, when the system is protected by mirror symmetry, these two bands will
unavoidably cross each other without opening a band gap.
In addition, a similar case also occurs in the D3. Therefore,
all Dirac cones in the β-boron sheet are protected by mirror
symmetry without exception.
To visualize the Dirac cone, we plot the 3D band structure
of D1 and D2 along -X in Figs. 3(a) and 3(c). The lower-
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XI ZUO et al.
PHYSICAL REVIEW B 100, 115423 (2019)
FIG. 4. The surface energy spectrum of β-borophene with
(a) armchair edge and (b) zigzag edge. The light blue color represents
contributions from the bulk atoms, while the white-red line stands for
contributions from the edge atoms. The nontrivial states are indicated
by the yellow arrows.
FIG. 3. (a), (c) 3D band structure around D1 and D2. (b),
(d) Charge densities for the states of the lower-energy band (LB) and
higher-energy band (HB) for D1 and D2, respectively.
energy and higher-energy bands of Dirac cone are denoted as
LB and HB, respectively. To take as an example, the energy
band of D1 has a larger Fermi velocity along the kx direction (v f = 1.33 × 106 m/s) than that along the ky direction
(v f = 0.35 × 106 m/s), which is comparable to the value
(v f = 0.82 × 106 m/s) in graphene. The anisotropic Dirac
cone with different Fermi velocities along different directions
suggests direction-dependent electronic structures of the βboron sheet. To unveil the origin for these anisotropic Dirac
cones, we calculate the band-decomposed charge density for
D1 and D2 in Figs. 3(b) and 3(d). For the LB state, the pz
orbitals of D1 and D2 form π band separated from each other.
However, for the HB state, the pz orbitals are localized on the
B dimers to form electron channels perpendicular to the LB
stripes. This indicates that the two Dirac cones are induced
by the weak interactions of the orthogonal-striped pz states,
which are protected by the mirror symmetry of My and Mx .
According to the bulk-boundary correspondence, the topologically nontrivial bulk-band topology is accompanied by the
existence of surface/edge states, which is usually detectable
from ARPES experiments. Based on the tight-binding Hamiltonian generated by MLWFs method, we calculate the edge
state of the β-boron sheet using the iterative Green’s function.
The edge spectra along the armchair and zigzag directions are
shown in Figs. 4(a) and 4(b), respectively. Along the armchair
direction, we can clearly see a nontrivial edge state (indicated
by the yellow arrow) connecting the nodal points of the nodal
lines. On the other hand, in the zigzag direction, one can also
find an edge state inside the two Dirac cones around D2 and
D3. The results provide further support for the existence of
nontrivial edge states of the nodal lines and Dirac fermions.
To further confirm the role of TRS and MRS in protecting
the closed nodal loop and open nodal line, we analyze the
effects of symmetry breaking on topological phases of the
β-borophene. On one side, we employ magnetic proximity
effect induced by the magnetic substrate. On the other side,
a vertical electric field is applied to induce Rashba SOC.
The nanodevice based on β-borophene in which the nodal
lines can be manipulated and controlled is shown in Fig. 5.
The device is composed of two electrical contacts (source
and drain), which sandwich the β-boron sheet grown on
YIG/SiO2 /Si substrate, and the gate voltage Vg is applied in
the normal direction of the sample surface. For this system,
the total Hamiltonian is composed of the Hamiltonian of
suspended β-borophene, the effective exchange interaction
term Hex , and Rashba SOC term Hrsb [see Eqs. (B1) and (B2)
in Appendix B].
The symmetry breaking induces a pronounced spin texture.
Since the electric field acts on a moving charge carrier as
an effective in-plane magnetic field, the Rashba SOC mixes
the spin-up and -down states, admitting a spin gap. Nevertheless, the out-of-plane exchange field shifts the Dirac cone
around D2 vertically, resulting in mixed “electron-hole” states
near the Dirac point. Since the Rashba SOC favors in-plane
alignment of electron spins and the exchange field tilts the
spins out of the plane, then the competition relation between
them leads to a noncoplanar band polarization. Figure 6
shows the evolution of the band structures of β-borophene
along the k path of −X --X with Rashba SOC λR and
exchange field Js . In the absence of exchange field and Rashba
SOC, there are two Dirac points and two inverted bands.
Owing to spin degeneracy, the crossing points are fourfold
FIG. 5. Schematic diagram of nanodevice based on β-borophene
grown on yttrium iron garnet (YIG) which produces magnetic proximity effects. Gate voltage Vg is applied to break mirror symmetry
of β-borophene, which induces Rashba spin-orbit interaction. The
vertical arrows indicate magnetization direction.
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PHYSICAL REVIEW B 100, 115423 (2019)
FIG. 6. Spin-resolved band structure along the high symmetry lines (-X --X) (a1) without exchange interaction or Rashba SOC term
(b1) with exchange interaction term (c1) with Rashba SOC term. The Js represents the effective interaction generated by the coupling of the
magnetic substrate with the electron spin. λR is the intensity of the Rashba term in eV. The 2D contours of the energy gap between the upper
and lower bands are shown in (a2), (b2), and (c2) in parallel with the band structure (a1), (b1), and (c1). Different colors in the left (right) panel
represent the expectation value of the spin of the bands (absolute value of energy gap between the bands). For example, purple (red) color in
the left panel corresponds to the spin-up (spin-down) state, Yellow color in the right panel denotes the location of zero-energy gap.
degenerate, while other points are doubly degenerate. Thus,
Dirac points and a perfect nodal loop can be found in
Fig. 6(a2), corresponding to the D2 and N2 in Fig. 2(a),
respectively. When the exchange magnetic field is applied perpendicular to the mirror plane, the exchange field breaks TRS
and lifts the spin degeneracy, leading to spin splitting [see the
violet (spin-up) and red (spin-down) curves in Fig. 6(b1)].
Because of the spin splitting, one crossing point in the -X
direction splits into four crossing points. Among them, two
are formed by the opposite spin states and the others are
created by the same spin states. Meanwhile, the initial Dirac
point splits into a pair of separated Weyl points [see zoom-in
band structure in Figs. 7(b1)–7(b3)]. Therefore, we obtain
two NLSMs formed by crossing between the states of the
opposite spins and two NLSMs stemmed from crossing of
the same spin states. For brevity, we only show the former
NLSMs in Fig. 6(b2) in two-dimensional MS. On the other
hand, if only Rashba SOC is included in Fig. 6(c1), the spinup and -down bands shift in opposite momentum directions
and create additional band-crossing points around N2 [see
zoom-in band structures in Figs. 8(c1)–8(c3)]. Therefore, we
can see in Fig. 6(c2) that two new nodal lines emerge around
the point, of which the shape is distinct from the original
nodal loop. Moreover, the Rashba SOC also mixes opposite
spin states and gives rise to anticrossing around the Dirac
cone (D2). In addition, when both TRS and MRS are broken,
Dirac points and part of the crossing points of the nodal
lines become gapped along -X [see Figs. 7(d1)–7(c3) and
Figs. 8(d1)–8(c3)]. As for open nodal line in the -Y highsymmetry line, the exchange magnetic field and Rashba SOC
induce similar spin splitting of the nodal lines (see Fig. 9).
Therefore, we acquire spin polarized open and closed nodal
lines in β borophene system by breaking TRS.
IV. CONCLUSIONS
We explore the nodal electronic phase of matter in emerging two-dimensional single-layer borophene based on firstprinciples calculations and tight-binding model. Aside from
the three usual Dirac cones close to the Fermi level like
graphene but with much higher Fermi velocity, we report
a coexistence of two types of nodal-line semimetal states,
namely, a spindlelike nodal loop and open nodal arc in the
recently synthesized β-boron sheet. To gain a deep insight into
the origin and characteristics of the nodal line, we carry out an
analysis through effective k · p model and make an inspection
of the edge states, we find that the nodal lines are protected by
time-reversal symmetry and mirror-reflection symmetry. By
breaking the time-reversal symmetry via magnetic proximity
effect, we find the original nodal lines evolve into a pair of
spin-polarized nodal loops concentrically distributed around
the point and open nodal arcs in the -Y direction. In addition, the inclusion of Rashba SOC breaks the mirror-reflection
symmetry and leads to the emergence of two new nodal lines
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XI ZUO et al.
PHYSICAL REVIEW B 100, 115423 (2019)
in 2D NLSMs systems, of which the shape is distinct from the
original nodal loop. These exotic properties make the β-boron
sheet as a promising candidate for the realization of future
high-speed electronic devices.
ACKNOWLEDGMENT
The authors thank financial supports from National Natural
Science Foundation of China (Grants No. 11574118 and
No. 21473102) and Natural Science Foundation of Shandong
Province (Grant No. ZR2019MA064).
APPENDIX A: k · p MODEL OF THE NODAL LOOP
Here, we discuss the nodal loop by deriving an effective
k · p model. In general, a two-band effective Hamiltonian can
be written as [12]
H (k) =
3
di (k)σi ,
(A1)
i=0
where di (k) (i = 0, 1, 2, 3) represents the real function of
k(kx , ky ). σ0 and σ1,2,3 are the 2 × 2 unit matrix and Pauli
matrices, respectively. The operators T̂ and M̂z corresponding
to TRS and MRS make the H (k) satisfy
T̂ H (k)T̂ −1 = H (−k),
M̂z H (k)M̂z−1 = H (k),
(A2)
where T̂ = K with K being the complex-conjugate operator
for the spinless case. M̂z can be replaced by σz due to the
opposite eigenvalues under the mirror-reflection symmetry.
By comparing the left side and right side of Eq. (A2), d1 (k) =
d2 (k) = 0, but d0,3 (k) are even functions of k. Then, we can
solve the secular equation and obtain the eigenvalue as
E (k) = d0 (k) ± |d3 (k)|.
(A3)
FIG. 7. Zoom-in band structure around the D2 point in
Fig. 2(a) in the presence of exchange field (Js )/Rashba spin-orbit
interaction (λR ). (a) The pristine band structure of β-borophene
around the D2 point. (b1)–(b3), (c1)–(c3), and (d1)–(d3) panels
display the evolution of band structure as Js , λR , and mixed Js and λR
increase, respectively. Colors indicate the expectation values of spin,
i.e., the curves in purple (red) correspond to the spin-up (spin-down)
state and the others denote spin-mixed states.
In addition, a vertical electric field is applied to induce Rashba
SOC. The schematic model is shown in Fig. 5. Hence, the total
Hamiltonian of the system is composed of the Hamiltonian
of suspended β-borophene, the effective exchange interaction
term Hex , and Rashba SOC term Hrsb . The Hex is described by
†
Hex = Jgs
CR,s,μ
[m̂.S]CR,s ,μ δμ,μ ,
(B1)
R,μ,μ ,s,s
where J is the effective exchange interaction generated by
the coupling of the magnetic substrate with the electronic
spin of the β-borophene, gs = 2 is the electron spin g factor, m̂ is the direction of the magnetization vector, while
The location of the band crossing depends on the form of
d3 (k). So here, we expand the d3 (k) to two orders
d3 (k) = a0 + a1 kx2 + a2 ky2 .
(A4)
At the band-crossing point, d3 (k) = 0. When a0 has an opposite sign with a1 and a2 , we obtain
ky2
kx2
+
−a0
−a0 = 1.
a1
a2
(A5)
This is actually the standard equation of the ellipse. Under
the condition of |a1 | > |a2 |, the trajectory of Eq. (A5) is
consistent with the shape of the nodal loop in the momentum
space, as shown in Fig. 2(c).
APPENDIX B: TIGHT-BINDING CALCULATION
The effects of symmetry breaking on topological phases of
the β-borophene on a magnetic substrate have been explored.
Our study focuses on the magnetic substrate which produces
the magnetization direction perpendicular to the sample plane.
FIG. 8. Zoom-in band structure around the N2 point in
Fig. 2(a) in the presence of exchange field (Js )/Rashba spin-orbit
interaction (λR ). (a) The pristine band structure of β-borophene
around the N2 point. (b1)–(b3), (c1)–(c3), and (d1)–(d3) panels
display the evolution of band structure as Js , λR , and mixed Js and λR
increase, respectively. Colors indicate the expectation values of spin,
i.e., the curves in purple (red) correspond to the spin-up (spin-down)
state and the others denote spin-mixed states.
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FULLY SPIN-POLARIZED OPEN AND CLOSED NODAL …
PHYSICAL REVIEW B 100, 115423 (2019)
S = (Sx , Sy , Sz ) are the Pauli spin matrices. Finally, CR,s,μ
denotes the annihilation operator which destroys an electron
in a spin s and angular momentum μ state at the unit cell
R. Equation (B1) describes the exchange interaction between
electron spin and the exchange field. The Hrsb is given by
Hrsb = iλr
†
{eik .r CR,s,μ
[σx r̂.ŷ − σy r̂.x̂] CR+r,s ,μ },
R,r,μ,s,s
FIG. 9. Zoom-in band structure along the -Y direction in the
presence of exchange field (Js )/Rashba spin-orbit interaction (λR ).
(a) The pristine band structure of β-borophene around the N1 point.
(b1), (b2) and (c1), (c2) panels display the evolution of band structure
as Js and λR increase, respectively. Colors indicate the expectation
values of spin, i.e., the curves in purple (red) correspond to the spinup (spin-down) state and the others denote spin-mixed states.
(B2)
where λr is the intensity of the Rashba term in eV, r corresponds to the hopping vectors between two sites (those vectors
consider atoms inside and outside the same unitary cell), x̂ and
ŷ are unit vectors along x and y axis, respectively. Because the
major contribution to the Rashba SOC is stemmed from the
nearest-neighbor hopping, we set a cutoff radius Rc = 7.8 Å
for our Rashba term, i.e., the Rashba terms with hopping
vectors larger than Rc are set to 0.
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