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Nano Res
DOI 10.1007/s12274-015-0842-7
Group 14 element based noncentrosymmetric quantum
spin Hall insulators with large bulk gap
Yandong Ma1 (), Liangzhi Kou2, Aijun Du3, and Thomas Heine1 ()
Nano Res., Just Accepted Manuscript • DOI 10.1007/s12274-015-0842-7
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1
TABLE OF CONTENTS (TOC)
Group 14 element based noncentrosymmetric quantum
spin Hall insulators with large bulk gap
Yandong Ma1*, Liangzhi Kou2, Aijun Du3, and Thomas
Heine1*
1Jacobs
University Bremen, Germany
2University
of New South Wales, Australia
3Queensland
University of Technology, Australia
Based on first-principles calculations, we predict a new family of 2D
inversion asymmetric TIs with sizeable bulk gaps in X2-GeSn (X=H,
F, Cl, Br, and I) monolayers.
Provide the authors’ webside if possible.
Author 3, http://staff.qut.edu.au/staff/du9/
Author 4, https://www.jacobs-university.de/ses/theine
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DOI (automatically inserted by the publisher)
Research Article
Group 14 element based noncentrosymmetric quantum
spin Hall insulators with large bulk gap
Yandong Ma1 (), Liangzhi Kou2, Aijun Du3, and Thomas Heine1 ()
Received: day month year
ABSTRACT
Revised: day month year
To date, a number of two-dimensional (2D) topological insulators (TIs) have
been realized in Group 14 elemental honeycomb lattice, but all are inversion
symmetric. Here, based on first-principles calculations, we predict a new family
of 2D inversion asymmetric TIs with sizeable bulk gaps from 105 meV to 284
meV, in X2-GeSn (X=H, F, Cl, Br, and I) monolayers, making them in principle
suitable for room-temperature applications. The nontrivial topological
characteristics of inverted band orders are identified in pristine X2-GeSn with
X=(F, Cl, Br, I), while for H2-GeSn, it undergoes a nontrivial band inversion at
8% lattice expansion. Topologically protected edge states are identified in
X2-GeSn with X=(F, Cl, Br, I) as well as in strained H2-GeSn. More importantly,
the edges of these systems, which exhibit single-Dirac-cone characteristics
located exactly in the middle of their bulk band gaps, are ideal for
dissipationless transport. Thus, Group 14 elemental honeycomb lattices make a
fascinating playground for manipulation of quantum states.
Accepted: day month year
(automatically inserted by
the publisher)
© Tsinghua University Press
and Springer-Verlag Berlin
Heidelberg 2014
KEYWORDS
Two-dimensional crystal,
Topological
insulators,
Dirac
states,
Band
inversion,
Strain
engineering, Group 14
honeycomb lattice.
1 Introduction
Topological insulators (TIs) represent a new state of
quantum matter that has generated great interest
within the condensed matter physics community
[1-6]. A TI is characterized by an insulating energy
band gap in the bulk and gapless boundary states. In
two-dimensional (2D) TIs, also known as quantum
spin Hall (QSH) insulators, the conducting edge
states
exhibit
dissipationless,
spin-polarized
conduction channels which are immune to
nonmagnetic scattering [7, 8]. Such properties make
Address correspondence to Y. Ma, [email protected]; T. Heine, [email protected]
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2D TIs better suited for low-power-consumption
electronics and error-tolerant quantum computing as
compared to three-dimensional (3D) TIs. Although
quite a few compounds, such as Bi2Se3, Bi2Te3, and
Sb2Te3, are well-established to be 3D TIs
experimentally [3, 9, 10], up to now there are only
two systems, namely the HgTe/CdTe [8] and
InAs/GaSb [11] quantum wells, that are
demonstrated to be 2D TIs in experiments. Even
worse is that the QSH effect in such two 2D TIs can
only be observed at ultralow temperature due to the
small bulk band gap of the order of meV. The need
for searching new QSH insulators with large bulk
band gaps is thus clear.
To date, extensive effort has been devoted to the
search for new QSH insulators with large bulk band
gap and stable structure. Several families of QSH
insulators, such as Bi/Sb honeycomb lattices [12-14],
bilayers of Group 13 elements with Bi [15],
ZrTe5/HfTe5 [16], 2D transition metal dichalcogenides
[17], and Group 14 honeycomb lattices [18-23], have
been proposed theoretically. Particularly, the Group
14 honeycomb lattice based QSH insulators are
expected to be more practical and promising. This is
largely thanks to their easy integration of TI states in
conventional electronic devices resulting from their
similarity with conventional semiconductors in
structures and chemical composition, as well as their
high thermal stability and their fascinating and
tunable electronic properties. Examples include
graphene [24], silicene [19], germanenen [19], stanene
[22], and their chemically modified counterparts [13,
14, 18, 20, 21, 23]. The Group 14 honeycomb lattices
could in principle provide a common host for many
topological phenomena and potential topological
applications. However, one obstacle is that the
realization of QSH insulator materials made of
Group 14 honeycomb lattices is limited to inversion
symmetric TIs, and for this materials class no
inversion asymmetric QSH insulator has ever been
observed or predicted so far. For TIs with inversion
asymmetry, they could not only preserve many
nontrivial
intriguing
phenomena
(i.e.,
crystalline-surface-dependent topological electronic
states [25, 26] and pyroelectricity [27]), but also offer
ideal
platforms
for
realizing
topological
magneto-electric effects [28, 29]. Therefore, one key
step toward future applications is to realize inversion
asymmetric TIs, especially those that are
electronically
stable
at
room
temperature.
Implementing those features in Group 14
honeycomb lattices would offer their integration
within the same honeycomb framework in Group 14
materials, such as germanene or stannene, and thus
potentially avoid contact resistance and other issues
of traditional device concepts.
Recently, Arguilla et al. [30] demonstrated that Sn
can be incorporated onto the 2D hydrogenated
germanene
(germanane):
they
successfully
synthesized
2D
Ge1−xSnxH1−x(OH)x
graphane
analogue from the topochemical deintercalation of
CaGe2–2xSn2x. Inspired by this experimental work,
here, we provide a systematical investigation on the
electronic and topological properties of free-standing
2D Group 14 honeycomb GeSn halide and GeSn
hydride (X2-GeSn, X=H, F, Cl, Br, and I) monolayers
(MLs) based on ab initio calculations. We predict that
X2-GeSn MLs with X=(F, Cl, Br, I) are large-gap 2D
TIs, presenting protected Dirac edge states that are
spin locked to momentum, and thus forming QSH
systems. On the other hand, H2-GeSn ML displays
normal band order but can be transformed into a
nontrivial topological phase with large energy gap
via appropriate strain engineering. Notably, the QSH
effect can be observed in an experimentally
accessible temperature regime in all these systems,
making them suitable for applications at
room-temperature. The nontrivial topological states
in these MLs are firmly established by the
identification of band inversion and the existence of
the topologically nontrivial edge states. Meanwhile,
all these 2D TIs reported here preserve inversion
asymmetry, making them unprecedented inversion
asymmetric polar QSH insulator candidates. The
present work not only expands the catalog of Group
14 honeycomb lattice based QSH insulators, but also
provides ideal platforms for realizing novel
topological phenomena.
2 Computational methods
First principles calculations are performed within the
density functional theory (DFT) using the projector
augmented wave method [31] as implemented in the
Vienna ab initio simulation package (VASP) [32, 33].
We employ the Heyd-Scuseria-Ernzerhof hybrid
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functional (HSE06) [34], which is well-known to
correctly predict the electronic structure. This
functional is computationally more involved than the
generalized
gradient
approximation
(GGA).
Therefore, for the calculation of edge states, being
computationally very involved, we have employed
the GGA, i.e. the Perdew−Burke−Ernzerhof (PBE)
functional [35]. For the interested theorist, a
comprehensive comparison between the results
obtained by PBE and HSE06 is presented in the
Supplementary Material. In short, while the PBE
electronic structure is generally in excellent
agreement with HSE06, as common in GGA the
conduction bands are shifted to lower energies and
consequently band inversions are predicted for
lattice elongations of 6% and higher. Strain is
simulated by changing the in-plane lattice constant a.
After applying strain, the atomic structure is
reoptimized. The magnitude of biaxial strain is
defined as ε = (a-a0)/a, where the lattice constant of
the strained and unstrained systems is equal to a and
a0, respectively. The plane-wave-cutoff energy is set
to 500 eV and the atoms are allowed to fully relax
until the force acting on each atom is less than 0.01
eV/Å . A vacuum space of no less than 18 Å between
neighboring systems is used. The Monkhorst and
Pack scheme [36] of k-point sampling is used for
integration over the first Brillouin zone. A 17×17×1
grid for k-point sampling is used for both geometry
optimization and self-consistent calculations.
second minimum with weaker corrugation (W) is
observed for a=4.41 Å . Such double-well energy
curve was also found in almost all of the previously
reported graphene-like systems [15, 38, 39]. The W
structure of H2-GeSn is more stable by 1.26 eV per
H2-GeSn formula unit. Therefore, in the remainder
we will focus on the more stable W structure. The
energy difference between H2-GeSn ML and its
homogeneous counterparts stannane and germanane,
defined as EH2-GeSn– 0.5(EH2-GeGe+EH2-SnSn), is only 32
meV per formula unit, indicating that H2-GeSn ML
could be stable. The stability of H2-GeSn ML is
further confirmed by the calculation of phonon
dispersion curves which are shown in Fig. 1(c) with
all vibrational modes having real frequencies.
3 Results and discussion
Figure 1 (a) Structural model (top and side views) of X2-GeSn
ML; the violet, green, and dark yellow balls indicate Ge, Sn, and
X=(H, F, Cl, Br, and I) atoms, respectively. (b) Energy versus
hexagonal lattice constant of X2-GeSn ML. (c) Phonon
dispersion curves for W H2-GeSn ML.
H2-GeSn ML is a strict graphane analogue, as
displayed in Fig. 1(a), where Ge and Sn atoms
exhibit sp3 hybridization which gives rise to the
buckled geometry as formed by the carbon atoms in
graphane [37]. The H-Ge bond is a little shorter than
that of H-Sn, which is rationalized by the different
atomic radii of the Group 14 elements and by the
difference in electronegativity. Fig. 1(b) shows the
variation of the total energy of H2-GeSn ML as a
function of the in-plane lattice constant a, where the
lattice constant a is varied to identify the structural
ground state. And the atomic positions are fully
relaxed at each fixed lattice constant. We observe two
energy minimums for H2-GeSn: At a=3.16 Å , a
strongly corrugated (S) minimum occurs, while a
The electronic band structure of H2-GeSn ML
without spin-orbit coupling (SOC) is depicted in Fig.
2(a). A direct band gap of 1.155 eV is predicted at the
Γ point. Away from Γ point, the conduction and
valence bands are well separated, we therefore only
focus on the bands around Γ point. By projecting the
bands onto different atomic orbitals we observe that
the conduction band minimum and the valence band
maximum at Γ point are mainly composed of one s
and two p orbitals, respectively, see Fig. 2(a). Here,
the two p orbitals are energy-degenerated. In trivial
insulators, such as graphane, the s orbital is typically
located above the two p orbitals in energy.
Consequently, H2-GeSn ML shows a semiconducting
nature with normal band order. When SOC is
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switched on, as shown in Fig. 2(f), the degeneracy of
p orbitals is lifted but the energy gap is maintained
although the gap size is reduced (0.977 eV). By
including SOC, no band inversion can be observed,
suggesting H2-GeSn ML is a normal insulator.
Figure 2 Energy band structure without (upper panels)
and with (lower panels) SOC for H2-GeSn ML with (a and
f) 0%, (b and g) 2%, (c and h) 4%, (d and i) 6%, and (e and j)
8% lattice expansion. Zero of energy is set at the Fermi
level. Insert: the main projection of the bands near the
Fermi level at Γ point.
External mechanical strain can reduce the band
gap size, which in turn, might introduce band
inversion even in systems with relatively weak SOC.
It can be seen clearly from Fig. 2 that the band
structure of H2-GeSn ML is indeed sensitive to the
lattice expansion and the gap values versus lattice
expansion are plotted in Fig. 3. From Fig. 2(a)-(e) and
Fig. 3 we can observe that, for H2-GeSn ML without
SOC, by increasing lattice expansion the conduction
band minimum at Γ point (that is the s state, as
labeled in Fig. 2) is gradually lowered in energy. This
yields a significant reduction of the global band gap
as well as the energy difference (Es-p) between the s
state and the upper p state at Γ point just around the
Fermi level. Without SOC, we do not observe a band
inversion in the band structure of H2-GeSn ML for
lattice expansions of up 8% [see Fig. 2(a)-(e)]. Thus,
the band inversion cannot be induced solely by
lattice expansion. Our phonon calculations suggest
that H2-SnGe ML can maintain its stability within
the strain range of 8% (see Fig. S1). By introducing
SOC, when the lattice expansion changes from 0% to
6%, the two p states still appear below the s state at
the Γ point, as shown in Fig. 2, indicating the
topologically trivial phase in H2-GeSn ML is still
unchanged. When the lattice expansion researches
8%, remarkably, one of the p orbitals shifted above
the s orbital. This suggests that the inversion of two
bands, which is a strong indication for the formation
of nontrivial topological states in H2-GeSn ML,
implying that H2-GeSn ML with 8% is a 2D TI.
Further proof, edge state calculations, will be shown
below. More importantly, the magnitude of the
nontrivial topological bulk band gap is 105 meV,
which is significantly larger than thermal
perturbation kBT at room temperature (26 meV). We
can therefore conclude that the QSH effect may be
measured in strained H2-GeSn ML at room
temperature.
Although the electronic and topological properties
of H2-GeSn ML are strongly sensitive to the lattice
expansion as well as to the choice of functional (see
Figure S2), the SOC-induced energy splitting (ESOC)
between the two p orbitals at Γ point around Fermi
level is not. This can be clearly seen in Fig. 3(a) and
(b) that ESOC remains unaffected with increasing
lattice expansion and is also irrespective of the choice
of PBE or HSE06 functional. These may be
understood by the fact that SOC strength is an
intrinsic property of the material and is hard to be
affected. Another interesting feature we can observe
is that ESOC based on PBE is almost similar to that
based on HSE06, see Fig. 3(a) and (b). This origin of
the similarity derives from that the ESOC is created by
SOC between the bonding and antibonding states of
the p orbitals in H2-GeSn ML which exhibits
relatively weak correlation effect.
X2-GeSn ML with X=(F, Cl, Br, I) shares a similar
geometric structure as H2-GeSn ML. As shown in Fig.
1(b), the variation of the total energy of X2-GeSn ML
with X=(F, Cl, Br, I) as a function of the hexagonal
lattice constant a also exhibits a double-well energy
curve, corresponding to S and W configurations.
And also for these MLs, the W structure is more
stable in energy and thus will be discussed in the
following. The energy difference between X2-GeSn
ML and its homogeneous counterparts, stannane and
germanane, is only 29, 22, 15, and 2 meV per formula
unit, respectively, indicating that X2-GeSn ML could
also be stable. From Fig. 1(b), we can observe that the
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lattice constant of X2-GeSn ML gradually decreases
with moving X from F to I; however, all of them are
significantly larger than that of the H2-GeSn ML. The
energy band structures of X2-GeSn ML with X=(F, Cl,
Br, I) are depicted in Fig. 4. In the case without SOC,
X2-GeSn ML with X=(F, Br, I) are semimetalic with
the valence band maximum and conduction band
minimum touching at the Γ point just at the Fermi
level, which indicates different electronic properties
from those of H2-GeSn ML. For all these four
systems, the states around the touching point are
dominated by the degenerated p orbitals.
Noteworthy, the s orbital at Γ point around the Fermi
level, which is normally located above the two p
orbitals, shifts downwards and lies below the two p
orbitals, implying the existence of nontrivial band
Figure 3 Global energy gaps with and without SOC, energy difference (Es-p) between the s and upper p orbitals at Γ point with and
without SOC, and the SOC strength (ESOC) as a function of lattice expansion for H2-GeSn ML calculated by (a) PBE and (b) HSE06
functional. (c) Global energy gaps of X2-GeSn ML with X=(F, Cl, Br, I). Insert in (a) and (c) plot the dipole moment of H2-GeSn versus
lattice expansion and the dipole moment of X2-GeSn versus X=(F, Cl, Br, I), respectively.
Figure 4 Energy band structure without (upper panels) and with
(lower panels) SOC for X2-GeSn ML with X=(F, Cl, Br, I). Zero
of energy is set at the Fermi level. Insert: the main projection of
the bands near the Fermi level at Γ point.
order at Γ point even without considering SOC.
However, for Cl2-GeSn ML, it preserves normal
band order, with the s orbital locating above the p
orbitals, instead of the nontrivial band order. The
band gap in Cl2-GeSn ML is so small that the band
order at Γ point for Cl2-GeSn ML can be inverted by
turning on SOC, leading to a nontrivial band order in
Cl2-GeSn ML. From Fig. 4, we can find that, after
turning on SOC, X2-GeSn MLs with X=(F, Cl, Br, I)
are in insulating phases with nontrivial band orders
at Γ point, which strongly suggests that X2-GeSn
MLs with X=(F, Cl, Br, I) are 2D TIs. This indicates
that, in contract to the case of H2-GeSn ML, X2-GeSn
ML with X=(F, Cl, Br, I) is a 2D TI without applying
any external strain. In fact, concerning the relation
between the lattice constant and the band structure,
we can find that X2-GeSn ML with X=(F, Cl, Br, I)
behaves like a tensile strained H2-GeSn ML.
Moreover, as indicated in Fig. 3(c), the inclusion of
SOC produces a topologically nontrivial band gap of
187, 124, 174, and 284 meV, respectively, for X2-GeSn
ML with X=(F, Cl, Br, I). Such large nontrivial band
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gaps are capable of stabilizing the boundary current
against the influence of thermally activated bulk
carriers, and thus are beneficial for high-temperature
applications.
Figure 5 The calculated edge states for (a) strained H2-GeSn ML
and (c) F2-GeSn ML. (b) The zigzag nanoribbon used in the
edge calculations. Zero of energy is set at the Fermi level.
The topologically nontrivial insulating nature in
X2-GeSn ML should support an odd number of
Dirac-like topologically protected edge states
connecting the valence and conduction bands. In
order to further confirm the topological phases of
these MLs, we calculate the edge states of X2-GeSn
ML by constructing a zigzag nanoribbon structure
based on a slab model (a 8% lattice expansion is
employed for H2-GeSn ML). Here, all the dangling
bonds at edge sites are passivated by hydrogen
atoms. Such a nanoribbon structure is asymmetric
with GeH and SnH chains terminated at each side.
(see Fig. 5(b)). The nanoribbon widths exceed 10 nm
to avoid the interaction between two edges. The
calculated electronic structures, for computational
reasons only at the less expensive PBE level, of
X2-GeSn MLs are presented in Fig. 5 and Fig. S5. It
can be seen clearly that two sets of conducting edge
states in the energy gap crossing linearly at Γ point,
reflecting the two opposite edges of the nanoribbon.
The edge states marked with red and blue lines
correspond to the top and bottom edges, respectively.
The appearance of such gapless helical edge states
inside the bulk band gap is consistent with the band
inversion identification, which further prove that
X2-GeSn ML with X=(F, Cl, Br, I) and the H2-GeSn
ML with 8% strain are indeed QSH insulators. It is
worth noting that each edge is supposed to possess
edge states with different band dispersions due to
their asymmetric nanoribbon structures. However,
there is no significant difference between the two
edge states in each X2-GeSn ML and the two Dirac
points in each system are all located exactly at the
Fermi level. Considering the large nontrivial bulk
energy gaps, such X2-GeSn ML might be ideal
platforms for realizing QSH effect and are highly
desirable for the applications of topological edge
states in electronic and spintronic devices.
Apart from the interesting topological properties
demonstrated in these MLs, remarkably, the valence
band maximum also experience a significant band
splitting and a slight shift off the Γ point as shown in
Fig. 4. It is well known that in systems with inversion
symmetry, all the energy states are spin-degenerated
as long as the time-reversal symmetry is held.
Breaking inversion symmetry lifts such degeneracy
and leaves the energy states being spin-splitted at
generic k points. Such characteristics in X2-GeSn ML
originate from the local dipole field which is induced
by its inversion asymmetric structure. In X2-GeSn
ML, the X, Ge, Sn and X layers stack alternatingly,
giving rise to the polarity along the stacking axis.
The corresponding dipole moments of X2-GeSn ML
are listed in the insert of Fig. 3. The existence of
dipole moments in X2-GeSn ML with X=(F, Cl, Br, I)
as well as H2-GeSn ML with 8% strain strongly
indicates that all these systems are inversion
asymmetric QSH insulators. It should be noted that
up to now, only strained HgTe has been
experimentally realized and proved to be a 2D TI
with inversion asymmetry, but displays a rather
small energy gap (~20 meV) [40]. While for all the
previously reported Group 14 honeycomb lattice
based TIs, they are all inversion symmetric, severely
limiting their realistic applications. With the
advancement in experiment techniques, several
possible routes reported in recent works [15, 41] are
expected to be viable for synthesizing GeSn ML.
With synthesized GeSn ML, X2-GeSn ML can be
manufactured by employing the method for
preparing hydrogenated graphene [42]. Furthermore,
for actual applications, one critical concern is the
interface effect on the topological properties. To this
end, one should look for a substrate with minimal
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interfacial interaction with X2-GeSn ML (such as the
passivated substrates), and a more comprehensive
study concerning these problems should be carried
out in further works.
4 Conclusions
In conclusion, we have investigated the electronic
and topological properties of X2-GeSn ML by means
of ab initio calculations. We find that X2-GeSn MLs
with X=(F, Cl, Br, I) are QSH insulators, while for
H2-GeSn ML a topological phase transition can be
found at a 8% lattice expansion. Remarkably, all
these systems display large nontrivial topological
band gaps, which potentially can support their
applications under room temperature. More
interestingly, X2-GeSn ML with X=(F, Cl, Br, I) and
H2-GeSn ML with 8% strain are all in inversion
asymmetric topological insulator phases, making
them ideal candidates for realizing new topological
phenomena. The artificial honeycomb lattices
reported here can be prototypes for future
exploration of new inversion asymmetric QSH
insulators with large nontrivial band gaps.
Acknowledgments
Financial support by the European Research Council
(ERC, StG 256962) is gratefully acknowledged.
Electronic Supplementary Material: Phonon
dispersion curves for strained H2-GeSn ML, energy
band structure for H2-GeSn ML with different lattice
expansions calculated with PBE, phonon dispersion
curves for X2-GeSn ML with X=(F, Cl, Br, I), energy
band structure for X2-GeSn ML with X=(F, Cl, Br, I)
calculated with PBE, and calculated edge states for
Cl2-GeSn, Br2-GeSn, and I2-GeSn MLs are available
in the online version of this article at
http://dx.doi.org/10.1007/s12274-***-****-*
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