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PAPER
Cite this: Phys. Chem. Chem. Phys.,
2017, 19, 2148
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Intrinsic magnetism and spontaneous band gap
opening in bilayer silicene and germanene†
Xinquan Wang and Zhigang Wu*
It has been long sought to create magnetism out of simple non-magnetic materials, such as silicon and
germanium. Here we show that intrinsic magnetism exists in bilayer silicene and germanene with no need to
cut, etch, or dope. Unlike bilayer graphene, strong covalent interlayer bonding formed in bilayer silicene and
germanene breaks the original p-bonding network of each layer, leaving the unbonded electrons unpaired
and localized to carry magnetic moments. These magnetic moments then couple ferromagnetically within
Received 20th October 2016,
Accepted 2nd December 2016
each layer while antiferromagnetically across two layers, giving rise to an infinite magnetic sheet with
DOI: 10.1039/c6cp07184h
fundamental band gaps of 0.55 eV and 0.32 eV for bilayer silicene and germanene, respectively. The
structural integrity and magnetic homogeneity. Furthermore, this unique magnetic ordering results in
integration of intrinsic magnetism and spontaneous band gap opening makes bilayer silicene and
www.rsc.org/pccp
germanene attractive for future nanoelectronics as well as spin-based computation and data storage.
Introduction
The pursuit of intrinsic magnetism within traditional nonmagnetic materials is among the broad effort to identify and
develop new systems in which the spin-polarized electrons can
be used to store and process information.1–3 It is well-known
that an individual atom with an odd number of electrons has a
magnetic moment from the unpaired electron(s), whereas materials
consisting of elements with an even number of electrons, including
carbon (C), oxygen (O), silicon (Si), and germanium (Ge), can also
possess unpaired electrons when covalent bonds are broken. For
instance, in defected or disordered group IV solids, a single
electron occupying defect levels can be detected in magnetic
resonance experiments, but the whole system lacks a magnetically ordered state due to the low concentrations of those defects
or disorders.4 Bonds can also be broken at the surfaces or
interfaces of solids, but covalent materials usually reconstruct
to eliminate dangling bonds or doubly fill them with electrons of
opposite spin. Graphitic C is an exception, where the magnetic
states are found to be located at the grain boundaries of
graphite2 and at the zigzag edges of graphene nanoribbons.5
However, intrinsic magnetism was believed to be impossible for
the other group IV elements such as Si and Ge because they lack
bulk graphitic phases.
Department of Physics, Colorado School of Mines, Golden, CO 80401, USA.
E-mail: [email protected]
† Electronic supplementary information (ESI) available: Details of all possible
bilayer silicene crystal structures and energetics, energy barriers for phase transition from the buckling structure to the planar structure, and phonon dispersion
and electronic band structure of bilayer germanene. See DOI: 10.1039/c6cp07184h
2148 | Phys. Chem. Chem. Phys., 2017, 19, 2148--2152
Fortunately, the rapid progress of modern nanotechnologies
has made it possible to realize graphitic Si in a two-dimensional
(2D) honeycomb structure via a bottom-up fabrication route: for
instance, the successful synthesis of 2D silicene on Ag(111),6–8
Ir(111)9 and ZrB2(0001)10 surfaces. The graphitic germanium,
germanene, is expected to be realized in the same manner.11 The
2D honeycomb phases of Si and Ge are analogues of graphene,12
and many intriguing properties of graphene such as Dirac
massless fermions and ultrahigh carrier mobility13 have also
been predicted in silicene and germanene.14,15 However, silicene
and germanene are buckled in each of their six-membered ring
units,15–19 instead of the flat 2D structure of graphene. This is
believed to be the result of the pseudo-Jahn–Teller distortion due
to the strong coupling between the occupied and unoccupied
molecular orbitals.
On the other hand, the stronger spin–orbit coupling enables
silicene and germanene to serve as potential 2D topological
insulators and exhibit the quantum Spin-Hall effect.20–24 Moreover, due to their compatibility with the current Si-based semiconductor technology, silicene and germanene have attracted
growing attention. However, a sizable band gap (Eg), which is
crucial for electronic and optical applications, does not yet exist
in free-standing silicene or germanene, though none-zero band
gaps were proposed with chemical functionalization.25
In this work our first-principles calculations reveal that both
intrinsic magnetism and band gap opening can be simultaneously achieved in the 2D bilayer silicene (germanene) system.
This is because of the strong covalent interlayer bonds established, which break the original p-bonding network within each
layer. An infinite magnetic sheet with structural integrity and
magnetic homogeneity thus is induced by the interplay between
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the intra-layer ferromagnetic and the interlayer antiferromagnetic
coupling. This is sharply different from the magnetism arising in
graphene nanoribbons where zigzag edges are required and the
magnetic moments are only locally distributed near edges. Such
magnetic ordering leads to band gaps of 0.55 eV and 0.32 eV for
bilayer silicene and germanene, respectively, in contrast to their
gapless non-magnetic phases.
Computational method
Our density-functional-theory (DFT) calculations are performed
using the plane wave VASP package.26 Since the interlayer
interaction is essential for determining the stacking configurations
for multilayers of graphene-like materials, and the conventional
generalized gradient approximations (GGA) cannot correctly
describe the weak dispersive interaction, we employ both the
PBE-GGA functional27 and the van der Waals density functional
denoted by optPBE-vdW28 to scrutinize the geometries and
binding energies between different stacking configurations.
The projected augmented wave (PAW) method29 is adopted to
describe the interactions between valence electrons and ions.
We have constructed 4 4 supercells consisting of 64 atoms
with an initial interlayer spacing of 8 Å to simulate all possible
stacking configurations. A sufficiently large vacuum depth of
35 Å is applied in order to eliminate the interactions between
periodic images. The reciprocal space is sampled with an
11 11 1 Monkhorst–Pack k-mesh, and the plane wave
cutoff energy is set to 450 eV. All structures are relaxed without
any symmetry constraints until the maximum atomic force is
less than 10 4 eV Å 1. We also performed the many-body GW
calculations30 using the plasmon-pole model of Hybertsen–
Louie31 to obtain accurate band gaps.
Results and discussion
Because of the buckling character, a silicene monolayer can be
divided into top and bottom sublattices, allowing the AA and
AB stacking of a bilayer system to have two and three variants
(ESI†), respectively. Our calculations using both PBE and
optPBE-vdW functionals indicate that the AB stacking with
the D3d point-group symmetry, as plotted in Fig. 1, is the most
energetically favored. We note that the optPBE-vdW functional
barely changes the interlayer distance relative to PBE, because
of the strong covalent interlayer bonds formed. The interlayer
bond length d2 = 2.524 Å, slightly longer than the intra-layer
bond length d1 = 2.323 Å; however, they are both longer than
2.271 Å for silicene monolayer and close to 2.368 Å for bulk Si,
suggesting a greater sp3 hybridization component for bilayer
silicene compared to the free-standing silicene monolayer.
Under compression, the two AA stacked configurations
can transform into a new planar phase with all Si atoms
fully bonded,19,32,33 with total energy lower than any buckled
configuration because of more interlayer bonds. However, such
phase transition needs to overcome considerable energy barriers
(about 0.2 eV per unit cell along both reaction paths, see ESI†);
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Fig. 1 Optimized geometric structure for bilayer silicene (germanene) of
AB stacking with interlayer bonds. (a) Top view. The rhombus marked by
blue dashed lines shows the unit cell. The lattice constant a = 3.854 (4.077) Å
for silicene (germanene). (b) Side view. There are two distinct Si–Si (Ge–Ge)
bond lengths: intra-layer d1 = 2.323 (2.475) Å and interlayer d2 = 2.524
(2.678) Å. Atoms belonging to different layers are distinguished by green and
red colors. Atoms at the S1 (S2) sites are bonded without (with) interlayer
bonds.
thus it cannot occur spontaneously under ambient conditions
because random thermal fluctuation is not able to activate
ordered phase transition for an infinite system. Therefore in
this work we only focus on the bonded AB-stacking configuration
(Fig. 1).
We also performed phonon-mode analysis to examine the
dynamical stability of bilayer silicene. Fig. 2 shows phonon
band structure for bilayer silicene, which has no imaginary
phonon modes. The highest optical branch is located at the
G-point, which belongs to a doubly degenerate mode with a
frequency of oG = 521 cm 1. This mode is Raman active and
nearly 10 cm 1 higher than the highest optical mode (510 cm 1
in LDA) in bulk silicon34 while 40 cm 1 lower than the highest
optical mode in the silicene monolayer,35 which again suggests
stronger sp3 hybridization in bilayer silicene than in the silicene
monolayer. The slopes of the longitudinal acoustic branches near
the G-point correspond to the speed of sound, indicating the
in-plane stiffness. Our results clearly show that the in-plane elastic
response of bilayer silicene is nearly isotropic with almost the same
sound speed, vs = 8.87 km s 1, along the G–M and G–K directions.
Fig. 2
Phonon dispersion in bilayer silicene.
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Such isotropic character in sound speed was also identified in the
silicene monolayer with vs = 9.49 km s 1,35 suggesting slightly
higher in-plane stiffness for the silicene monolayer than for bilayer
silicene.
To further verify thermodynamic stability, we also carried
out the Born–Oppenheimer molecular dynamics simulations
using a relatively large 10 10 supercell containing 400 Si
atoms. At room temperature the bilayer silicene keeps the AB
configuration after 5 ps (the time step is 1 fs) except for some
structural fluctuations. Though 5 ps is not long enough to
demonstrate its structural stability, together with phonon
spectrum we expect the proposed bilayer silicene to be quite
stable at room temperature.
Within the silicene monolayer, the overlap of pz orbitals
between neighboring Si atoms is much smaller than that of C
atoms in graphene, resulting in the sp3-like hybridization
(instead of sp2) by buckling to gain stability: 3 of the four
sp3-like orbitals form covalent bonds with neighboring atoms
while the last one with upward spin interacts with its adjacent
downward counterparts, leading to an extensive p-bond network
in the nonmagnetic ground state.36 But the formation of interlayer bonds in bilayer silicene will destroy the p-bond network of
each layer, and the electron in the pz orbital with no interlayer
bonds (at the S1 sites indicated in Fig. 1) thus becomes unpaired
and localized to carry a magnetic moment of B1 mB.
To study the preferred coupling among these magnetic
moments, we have examined all possible magnetic configurations within a 2 2 supercell. In general, on applying any
antiferromagnetic ordering to one layer as the initial state, the
system would spontaneously decay into a non-spin-polarized
state upon optimization. In contrast, the initial ferromagnetic
ordering within a silicene layer is well preserved, and the
system is further stabilized by the interlayer antiferromagnetic
coupling, whereas ferromagnetic coupling across two layers will
also lead to a non-spin-polarized state.
Such magnetic ordering lowers the total energy by 10 meV
per unit cell against the non-spin-polarized state. According
to the mean field theory, the Curie temperature (Tc) can be
estimated by
1
2gkBTc
= ENM
EAFM,
(1)
where g is the system dimension, kB the Boltzmann constant,
and ENM and EAFM the total energies per unit cell for the nonmagnetic and the antiferromagnetic state, respectively. Here
g = 2 for bilayer silicene, and thus Tc is estimated to be 116 K.
We note that this simple estimation is based on the classical
Heisenberg model, and it tends to overestimate Tc compared
with more rigorous approaches.
Fig. 3 shows spin distributions in the bilayer silicene, which
demonstrates that the magnetic moments are mainly localized
at the S1 sites, while the S2-site atoms carry much smaller local
magnetic moments around 0.10 mB. The Si atoms at the S2 sites
are diamagnetic due to their fully filled sp3 orbitals, whose
magnetization is induced by the local magnetic field created
by neighboring magnetic moments at the S1 sites, so that the
2150 | Phys. Chem. Chem. Phys., 2017, 19, 2148--2152
Fig. 3 (a) Top view and (b) side view for the calculated spin density (ra rb)
of bilayer silicene, respectively. The spin-up density is in red while the spindown density is in green, and the isosurfaces are plotted at 0.25 electrons
per Å3. (c) The projected electron DOS of a Si atom at the S2 site of the
bottom layer.
induced moment at each S2 site is of opposite orientation to its
neighboring moments at S1 sites.
To explain why ferromagnetic coupling within each layer is
favored over the non-spin-polarized interaction, we analyze the
electron density of state (DOS) projected on a S2 site in the
bottom layer. As shown in Fig. 3c, there is no exchange splitting
between the two spin channels in the non-spin-polarized state.
However, upon spin-polarization, discrepancy is observed near
the Fermi level between the two spin channels, where the spindown states have been partially drained above the Fermi level,
meanwhile the same amount of spin-up states above the Fermi
level have been pulled down and become occupied. The
exchange splitting results in net magnetic moment, and it also
pushes down the valence band maximum (VBM) to 0.17 eV below
Fermi level (under PBE), suggesting a lowered total energy. The
magnetic coupling between S1 moments across two layers prefers
antiferromagnetism, and the induced S2 moments couple antiferromagnetically between two layers as well, which in turn
mutually enhance the exchange splitting between each other
due to their diamagnetic nature.
Magnetism is normally attributed to d- or f-electrons, but
present results point to the possibility of intrinsic magnetism
arising from 2D Si crystals without any impurity. The valence
electrons derived from the p-states are much more delocalized
than d- or f-electrons; thus they can promote the long-range
exchange interaction over much larger spatial extension. Consequently, the magnetic coupling between 3p moments explored
in bilayer silicene can be attributed to the much extended p–p
interaction mediated by their wave function tails.
Intrinsic magnetism has been revealed in graphitic C
materials,2,5 but the resultant magnetism is mainly distributed
along the zigzag edges. Unlike graphene, magnetism in bilayer
silicene is caused by the collapse of the p-bond network rather
than edge states; thus it is not necessary to cut or etch out
certain edges and the integrity of silicene structure is well
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Fig. 4 (a) DFT electronic band structure for bilayer silicene. The two
bands in black near the Fermi level are those for the non-spin-polarized
state. (b) The PDOS projected on a Si atom at the S1 site (top panel) and S2
site (bottom panel), respectively.
preserved. The induced magnetism is uniformly distributed all
over the 2D crystal. Recently, synthesis of silicene on various
substrates has been reported, but no free-standing silicene
monolayer has been exfoliated from any substrates yet.6–10
Due to strong interaction with the substrate, the synthesized silicene
monolayer has a buckled structure with an undulation,7–10 which
might hinder the formation of ordered interlayer bonds when
the second Si layer is deposited. The bilayer structure we
studied is expected to be formed experimentally by stacking
two free-standing silicene monolayers.
The electronic properties of bilayer silicene have been
modeled by tight-binding (TB) approximations based on p-states
of silicon, which predicts the bilayer silicene to be metallic with a
0.3 eV overlap between the valence and conduction bands.37 But
the TB model ignores the crucial spin degree of freedom. Fig. 4
shows DFT electronic band structures of bilayer silicene; without
spin-polarization, bilayer silicene is metallic, in agreement with
the aforementioned TB model. However, upon including the spin
degree of freedom, an indirect band gap of 0.29 eV appears, with
the VBM at the K-point and the conduction band minimum
(CBM) along the K–G direction.
To overcome the well-known problem of severe underestimation of band gaps by DFT, we performed self-energy calculations employing the many-body GW approach. The band gap
increases to 0.55 eV, whereas the electronic structures of the
valence and conduction bands are close to those obtained by
PBE. There have been attempts to substitute Si metal-oxidesemiconductor field effect transistors (MOSFETs) with grapheneor silicene-based materials owing to their extremely high carrier
mobility,38 but a sizable band gap larger than 0.4 eV is critical for
this purpose. The 0.55 eV band gap predicted in bilayer silicene
is large enough to pave the way for high-performance FET
fabrication with silicene.
The band gap in bilayer silicene originates from the interlayer
antiferromagnetic superexchange interaction. Similar antiferromagnetic inter-edge interaction has already been identified as the
origin of band gap opening in graphene zigzag nanoribbons.39,40
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Here the magnetism is mainly contributed by the 3pz orbitals,
which is demonstrated by analyzing the partial density of states
(PDOS) of each atom. As summarized in Fig. 4b, for Si atoms at
both S1 and S2 sites, their PDOS are all rearranged relative to those
of the non-spin-polarized state to assure exchange splitting
between different spin channels, and the pz DOS have the largest
mismatch. The exchange splitting suppresses the VBM by 0.23 eV
and shifts the CBM upward by 0.37 eV relative to those of
non-spin-polarized state, leading to a PBE band gap of 0.29 eV.
Since the Si atoms share the same PDOS distributions as their
counterparts on the other layer except for switching the spin
channels, the system has a zero total magnetic moment. We
note that previous theoretical studies25,41 have demonstrated
magnetic properties of silicene by functionalization of bromine
atoms or by hydrogenation; here we find intrinsic antiferromagnetism in silicene without any chemical manipulation of the
surface. Experimentally, Yaokawa et al.42 have successfully synthesized three types of bilayer silicenes using calcium-intercalated
monolayer silicene (CaSi2) with a BF4 -based ionic liquid.
We also studied bilayer germanene, which can form the
configuration of AB stacking with interlayer bonds as well. Its
dynamical stability is demonstrated by phonon-mode analysis
(Fig. 3S(a) in ESI†). There are two distinct Ge–Ge bonds: d1 = 2.475 Å
within each layer and d2 = 2.678 Å between two layers, and both are
longer than the uniform bond length of 2.442 Å for the germanene
monolayer, suggesting stronger sp3 hybridization in bilayer
germanene than in the germanene monolayer. Furthermore,
the same magnetic ordering in bilayer germanene as that in
bilayer silicene is identified, which lowers the total energy by
2.8 meV per unit cell relative to that of the non-spin-polarized
state. The spin-polarization leads to a spontaneous band gap
opening of 0.13 eV by the PBE functional (Fig. 3S(b) in ESI†), and
Eg = 0.32 eV using the GW approximation. However, in bilayer
germanene the VBM moves to the G point rather than staying
along the K–G direction, whereas the CBM is still located near
the K point. The Curie temperature is estimated to be 33 K, much
lower than that of bilayer silicene.
Conclusions
In summary, our first-principles analysis reveals the intrinsic magnetism and spontaneous band gap opening in bilayer silicene and
germanene. The magnetic moments in Si or Ge atoms couple
ferromagnetically within each layer while antiferromagnetically
across two layers, resulting in reduced total energy and a transition
from metal to semiconductor. Although the Curie temperatures are
much lower than room temperature, the integrity of bilayer structure, homogeneity of magnetism, substantial band gap, and compatibility with Si-based technology make these materials very appealing
to experimental efforts on realization and characterization.
Acknowledgements
This work was financially supported by DOE Early Career Award
(No. DE-SC0006433). Computations were carried out at the
Phys. Chem. Chem. Phys., 2017, 19, 2148--2152 | 2151
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Golden Energy Computing Organization (GECO) at the CSM
and National Energy Research Scientific Computing Center
(NERSC).
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References
1 P. Esquinazi, D. Spemann, R. Höhne, A. Setzer, K.-H. Han
and T. Butz, Phys. Rev. Lett., 2003, 91, 227201.
2 J. Červenka, M. I. Katsnelson and C. F. J. Flipse, Nat. Phys.,
2009, 5, 840.
3 S. C. Erwin and F. J. Himpsel, Nat. Commun., 2010, 1, 58.
4 E. L. Elkin and G. D. Watkins, Phys. Rev., 1968, 174, 881–897.
5 H. Lee, Y.-W. Son, N. Park, S. Han and J. Yu, Phys. Rev. B:
Condens. Matter Mater. Phys., 2005, 72, 174431.
6 P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis,
M. C. Asensio, A. Resta, B. Ealet and G. Le Lay, Phys. Rev. Lett.,
2012, 108, 155501.
7 B. Feng, Z. Ding, S. Meng, Y. Yao, X. He, P. Cheng, L. Chen
and K. Wu, Nano Lett., 2012, 12, 3507–3511.
8 R. Arafune, C.-L. Lin, K. Kawahara, N. Tsukahara, E. Minamitani,
Y. Kim, N. Takagi and M. Kawai, Surf. Sci., 2013, 608, 297–300.
9 L. Meng, Y. Wang, L. Zhang, S. Du, R. Wu, L. Li, Y. Zhang, G. Li,
H. Zhou, W. A. Hofer and H.-J. Gao, Nano Lett., 2013, 13, 685–690.
10 A. Fleurence, R. Friedlein, T. Ozaki, H. Kawai, Y. Wang and
Y. Yamada-Takamura, Phys. Rev. Lett., 2012, 108, 245501.
11 E. Bianco, S. Butler, S. Jiang, O. D. Restrepo, W. Windl and
J. E. Goldberger, ACS Nano, 2013, 7, 4414–4421.
12 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,
Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov,
Science, 2004, 306, 666–669.
13 A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov
and A. K. Geim, Rev. Mod. Phys., 2009, 81, 109–162.
14 L. Chen, C.-C. Liu, B. Feng, X. He, P. Cheng, Z. Ding, S. Meng,
Y. Yao and K. Wu, Phys. Rev. Lett., 2012, 109, 056804.
15 D. Jose and A. Datta, Acc. Chem. Res., 2014, 47, 593–602.
16 D. Jose and A. Datta, J. Phys. Chem. C, 2012, 116, 24639–24648.
17 D. Jose, A. Nijamudheen and A. Datta, Phys. Chem. Chem.
Phys., 2013, 15, 8700–8704.
18 A. Nijamudheen, R. Bhattacharjee, S. Choudhury and
A. Datta, J. Phys. Chem. C, 2015, 119, 3802–3809.
19 J. E. Padilha and R. B. Pontes, J. Phys. Chem. C, 2015, 119,
3818–3825.
20 M. Ezawa, Phys. Rev. Lett., 2012, 109, 055502.
2152 | Phys. Chem. Chem. Phys., 2017, 19, 2148--2152
Paper
21 C.-C. Liu, W. Feng and Y. Yao, Phys. Rev. Lett., 2011, 107, 076802.
22 R.-w. Zhang, C.-w. Zhang, W.-x. Ji, P. Li, P.-j. Wang, S.-s. Li
and S.-s. Yan, Appl. Phys. Lett., 2016, 109, 182109.
23 R.-w. Zhang, W.-x. Ji, C.-w. Zhang, S.-s. Li, P. Li and P.-j.
Wang, J. Mater. Chem. C, 2016, 4, 2088–2094.
24 C. Huang, J. Zhou, H. Wu, K. Deng, P. Jena and E. Kan,
J. Phys. Chem. Lett., 2016, 7, 1919–1924.
25 F.-b. Zheng and C.-w. Zhang, Nanoscale Res. Lett., 2012, 7, 422.
26 G. Kresse and J. Furthmüller, Comput. Mater. Sci., 1996, 6,
15–50.
27 J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett.,
1996, 77, 3865–3868.
28 J. c. v. Klimeš, D. R. Bowler and A. Michaelides, Phys. Rev. B:
Condens. Matter Mater. Phys., 2011, 83, 195131.
29 P. E. Blöchl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994,
50, 17953–17979.
30 G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys., 2002, 74,
601–659.
31 M. S. Hybertsen and S. G. Louie, Phys. Rev. B: Condens.
Matter Mater. Phys., 1986, 34, 5390–5413.
32 J. Bai, H. Tanaka and X. C. Zeng, Nanoscale Res. Lett., 2010,
3, 694–700.
33 D. Jose and A. Datta, Phys. Chem. Chem. Phys., 2011, 13,
7304–7311.
34 S. Wei and M. Y. Chou, Phys. Rev. B: Condens. Matter Mater.
Phys., 1994, 50, 2221–2226.
35 J.-A. Yan, R. Stein, D. M. Schaefer, X.-Q. Wang and M. Y.
Chou, Phys. Rev. B: Condens. Matter Mater. Phys., 2013,
88, 121403.
36 G. G. Guzmán-Verri and L. C. Lew Yan Voon, Phys. Rev. B:
Condens. Matter Mater. Phys., 2007, 76, 075131.
37 F. Liu, C.-C. Liu, K. Wu, F. Yang and Y. Yao, Phys. Rev. Lett.,
2013, 111, 066804.
38 F. Schwierz, Nat. Nanotechnol., 2010, 5, 487.
39 Y.-W. Son, M. L. Cohen and S. G. Louie, Phys. Rev. Lett.,
2006, 97, 216803.
40 J. Jung, T. Pereg-Barnea and A. H. MacDonald, Phys. Rev.
Lett., 2009, 102, 227205.
41 R.-w. Zhang, C.-w. Zhang, W.-x. Ji, S.-j. Hu, S.-s. Yan, S.-s. Li,
P. Li, P.-j. Wang and Y.-s. Liu, J. Phys. Chem. C, 2014, 118,
25278–25283.
42 R. Yaokawa, T. Ohsuna, T. Morishita, Y. Hayasaka, M. J. S.
Spencer and H. Nakano, Nat. Commun., 2016, 7, 10657.
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