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PCCP
Physical Chemistry Chemical Physics
Accepted Manuscript
This article can be cited before page numbers have been issued, to do this please use: A. Ebrahimian and
M. Dadsetani, Phys. Chem. Chem. Phys., 2017, DOI: 10.1039/C7CP05844F.
Volume 18 Number 1 7 January 2016 Pages 1?636
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Page 1 of 25
Physical Chemistry Chemical Physics
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DOI: 10.1039/C7CP05844F
groups in Two-dimensional Molybdenum and Tungsten Dinitride
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Ali Ebrahimian?; Mehrdad Dadsetani*
Department of Physics, Lorestan University, Khoramabad, Iran
Using the ab initio methods we have investigated the topological and optical properties of the
surface functionalized XN2 sheets (X=Mo, W). On the basis of first principles calculations and
K.p effective model, we report the existence of topological nodal-line states in the Potassium
functionalized XN2 sheets (K2MoN2 and K2WN2). We show that a nodal line ring exists near the
Fermi level in the absence of spin-orbit coupling (SOC). When SOC is included, the bandcrossing points are gap out giving rise a new nodal ring along
?-K.
These band-crossing is
protected by the existence of mirror reflection and time-reversal symmetry. Our calculations show
that the inclusion of electron-hole (e-h) interactions in the calculations of optical absorption of
functionalized MoN2, reveal the presence of strongly bound excitons below the absorption onset
where the number of bound excitons depend strongly on the terminated surface group. Moreover,
the surface terminated groups strongly change the energy distribution range of exciton which can
be used to tune the optical absorption at infrared(IR) and visible light. Interestingly the F2MoN2
have several strongly bound excitons with binding energy of 1.35 eV for the first exciton which is
larger than corresponding one in transition metal dichalcogenide MoS2.
Keywords: Topological semimetal, Nodal-line states, Band inversion, Excitonic effects.
*Corresponding author; Tel/ fax: +986633120611.
E-mail address: dadsetani.m@lu.ac.ir
? Email: ebrahimian.Al@fs.lu.ac.ir
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Physical Chemistry Chemical Physics Accepted Manuscript
Dependence of Topological and Optical Properties on the Surface terminated
Physical Chemistry Chemical Physics
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DOI: 10.1039/C7CP05844F
I. INTRODUCTION
Recent theoretical and experimental studies reveal different classes of topological phases,
[3]. The experimental discovery of Dirac [4] and Weyl [5] semimetals and their unique
topological properties has attracted great research interest in topological semimetals. Despite
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being a gapless metal, a topological semimetal is characterized by topological invariants,
broadening the classification of topological phases of matter beyond insulators. In contrast to
topological insulators where only surface states are interesting[6], a topological semimetal
features an unusual band structure in the bulk and on the surface.
In 3D topological semimetals, the conduction and the valence bands have robust crossing points
which result in new types of Fermi surfaces composing of points or lines. The dimension and the
degeneracy of band-crossing can be used to classify topological semimetals including Dirac,
Weyl and nodal line semimetals. In Dirac semimetals, energy bands are twofold degenerate and
conduction bands touch the valence bands at certain momentum points so the band-crossing
points are 0-dimensional (0D) and fourfold degenerate. These 0D band touching points become
doubly degenerate with definite chirality in Weyl semimetals. In fact, Dirac point is composed of
two Weyl points which can be separated by breaking time reversal (TR) or inverssion symmetry.
A nodal-line semimetals (NLSM) has extended band crossing along a continuous one dimensional
curve in K space which is topologically protected by extra crystalline symmetries such as mirror
reflection [7, 8] and glide plane [9, 12]. In other words, a band crossing forms a stable line node
against band hybridization if it locates on a high symmetry plane and each energy band belongs to
different eigenstates of crytalline symmetry. The nodal-line induces drumhead-like surface states
while Weyl nodal points are connected on the boundary by Fermi arc surface states. So far, most
of the reported nodal-line semimetals are 3D materials[13, 23]. Recently nodal-line semimetals
were extended to the symmetry-protected 2D materilas[24] and several 2D nodal-line semimetals
have been theoretically proposed, including a family group of MX ( M=Pd, Pt; X=S, Se, Te) [25],
mixed lattices composed of kagome and honeycomb lattices [26], two-dimentional compounds
X2Y (X=Ca, Sr and Ba, Y=As, Sb, Bi)[27] and bipartite square lattices[28]. However, this
expanding class of 2D topological semimetals needs more research to be understood
comprehensively. Therefore, finding new 2D topological nodal-line semimetal can improve our
knowledge of topological semimetals and their properties.
It is well known that surface termination on 2D materials may drastically change the topological,
electronic and optical properties. Indeed, it has been reported that surface termination induces
topological phase in 2D materials result in new topological materials. Recent first-principles
calculations have predicted the existence of quantum spin Hall (QSH) effect in hyrogenated XN2
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Physical Chemistry Chemical Physics Accepted Manuscript
including topological insulators (TIs) [1], topological semimetals(TSMs)[2] and superconductors
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Physical Chemistry Chemical Physics
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DOI: 10.1039/C7CP05844F
(H2MoN2 and H2WN2) [29]. Furthermore, Y. Ding et al. [30] have reported that F2MoN2 and
Cl2MoN2 sheets are semiconductors with indirect band gaps while the alkali functionalized ones
groups can be used to induce a wide range of band gap values in XN2 ( Mo, W) sheets promising
for optical applications with optical tunability over a large spectral range. Therefore, it is
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considerably interesting to explore the optical properties of functionalized XN2 (Mo and W)
sheets. As far as we know there is no report concerning the effect of surface terminated groups on
the optical properties of XN2 ( Mo, W) sheets. Due to large effective masses of the electrons and
holes and weak screening of Coulomb interaction in 2D systems, excitons (electron-hole pairs)
have main role in the optical properties of functionalized XN2 ( Mo, W) sheets [31]. Having
similar structures, Electron-hole pairs with large binding energy of over 0.5 eV have been
reported in transition metal dichalcogenide (TMD) monolayers [32,33,34]. Such strongly bond
excitons can possess large radii, as seen in MoS2 and MoSe2[32,34]. Indeed, the absorbance
spectra of these TMDs are characterized by the presence of two-exciton peaks. So in analogy to
MoS2 as a transition metal dichalcogenide (TMD), the investigation of the optical properties of
functionalized XN2 (Mo and W) sheets is important for development of the semiconductor
materials science. Motivated by these reports and the powerful ability of density functional
theory(DFT) in prediction of unusual properties of matter, we have investigated the structural and
optical properties of functionalized XN2 (Mo and W) sheets with alkali metal atoms and explored
the possible exsitence of topological phases in these 2D materials. The XN2 (X=Mo, W) has the
same structure as MoS2 in which two layer of Nitrogen atoms in a 2D hexagonal lattice stacked
along z axis to make trigonal prisms containing X atoms. The H-type MoN2 can be found in the
form of ternary AMoN2 (A=Li, Mg, Ca) bulk compounds in which H-MoN2 layers are separated
by intercalated alkali metal atoms [35]. Moreover, recently layered MoN2 materials have been
successfully synthesized utilizing a solid-state ion-exchange process. The Nitrogen atoms on the
surface of 2D XN2 are active [36], so its surfaces are usually termintated by some atoms suach as
Hydrogen (H), halogen atoms (F, Cl, Br) and alkali atoms (Li,Na, K)[37, 30]. As mentioned
before it has been reported [30] that the surface decorations on XN2 sheets could stabilize its
structure and change its electronic properties causing the appearance of new phenamena. In this
work, based on first-principles calculations and effective K.p model analysis, we report the
existence of topological nodal-lines in the compound K2XN2 (X=Mo, W) where nodal-lines are
protected by mirror reflection symmetry of the space group ( ? ?). The band inversion
happenes between the X-d orbital and the N-p (K-s) orbital. The results are robust against the use
of different exchange-correlation functional approximations. Indeed, to avoid the possible
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Physical Chemistry Chemical Physics Accepted Manuscript
can be semiconducting, semimetallic or metallic systems. This means that the surface terminated
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underestimation of the fundamental gap, we have checked the band-crossing by using the hybrid
functional HSE06 [38, 39] as well as the modified Becke-Johnson (mBJ) functional[40].
the optical properties of functionalized MoN2 sheets. To study the quasiparticle energy and
optical excitations of functionalized MoN2 sheets, we apply the first principle Bethe?Salpeter
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equation (BSE) approach[41, 42]. Our calculations show that there are bound excitons in the
absorbance spectra of functionalized MoN2 sheets and more interestingly the exciton binding
energies depend on surface terminated group. In the next section, We describe the theoretical
framework within which the results have been obtained. Section 3 is devoted to the discussion of
the results of study concerning the structural, electronic and optical properties of the XN2 sheets
(X=Mo, W). Finally we summarize our calculation results.
II.CALCULATION METHOD
For high accuracy electronic structure calculations, the structural optimization and electronic
properties were carried out using the all-electron full potential linearized augmented plane wave
(FP-LAPW) method as implemented in WIEN2k code [43] and FHI-aims code package which is
an all-electron full potential electronic structure code including numerical atom-centered orbitals
[44]. The exchange-correlation potential was treated within the generalized gradient
approximation of Perdew-Burke-Ernzerhof (PBE) [45]. The K-point sampling grid in the
electronic properties calculations was 60�� All the structures have been fully relaxed until
the maximum residual force on each atom becomes less than 10-4eV/ �. The spin-orbit coupling
(SOC) is included self-consistently within the second-variational method. In order to check the
possible underestimation of band gap within PBE functional, we have also employed the hybrid
density functional(HSE06) [38, 39] and the mBJ potential [40]. Recent first principle calculations
on the electronics structures of large class of materials including metal oxides and chalcogennides
show that HSE06 XC functional (Heyd-Scuseria-Ernzerhof) and the modified Becke-Johnson
(mBJ) functional are very successful in electronic structure calculations [46].
A 19 � thick vacuum layer is used to avoid interactions between nearest layers.
The Bethe-Salpeter Equation for two-particles Green's function [41, 42] is solved using the
Exciting code [47]. The matrix eigenvalue form of BSE is given by [41, 48-49]
?
,
= (1)
The indices() and k stand for valence (conduction) band and vector k in the irreducible part of
the Brillouin zone. Eigenvalues and eigenvectors represent the excitation energy of the
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Physical Chemistry Chemical Physics Accepted Manuscript
As the surface functionalized MoN2 and WN2 have similar band structures, we will only consider
Page 5 of 25
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jth correlated e-h pair and the coupling coefficient used to construct exciton wave function,
respectively. describes all interaction in the optical processes, consists of three interaction
elements. The last two terms are responsible for the formation of bound excitons. Using
eigenvalues and eigenvectors of BSE, the long wavelength limit of the imaginary part of the
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dielectric function () is given [42]
(?) =
!
?$ "?
A$
%|'( |
) /
*+, -*.,
" � ?(E$ ? ?)(2)
where ? and ? stand for the crystal volume and the frequency respectively. %4|P6 |4) is the
optical matrix element of momentum operator. The valence and conduction state energies and
are approximated by Kohn-Sham eigenvalues. To study the convergence of the excitonic
features in the optical properties, for all structures, the dielectric function has been calculated for
different numbers of K-point, valence states and conduction states. The momentum matrix
elements for the optical properties have been converged using 14��K-points. The excitonic
effects in the dielectric function of functionalized MoN2 sheets were converged by including 10
valence and 20 conduction states.
III. RESULTS AND DISCUSSION
1. STRUCTURAL AND ELECTRONIC PROPERTIES
The crystal structure of alkali metal functionalized XN2 is shown in Fig. 1. As it was shown in
previous ab initio calculations [30, 50, 51] the pristine XN2 sheet has H- and T-type geometrical
structures. Pristine XN2 is a three-layer structure with trigonal lattice. The X (Mo, W) atoms are
sandwiched between two N layers. These active layers of N atoms can be terminated by alkali
metal (Li, Na and K) or halogen atoms (Cl, F) to tune the electronic properties of XN2 [30]. As
shown in Fig. 1, for each phase of XN2(H- or T-type), the alkali (halogen) atom can locate on the
top of the N atoms (A), the X atoms (B) or the hollow sites (C). Our calculations show that the Csite alkali metal functionalization of the H-type obtain the lowest energy and is more stable than
other configurations. For halogen atom functionalized XN2, the most stable configuration is the
H-type where the chemical groups are located on the top of the A sites (N atoms). These results
are in good accordance with previous report [30]. After surface functionalization with halogen
atoms (F, Cl) the X2MoN2 sheet become semiconductor where its band gap depends on
terminated group. Our calculations show that Na2MoN2 and K2MoN2 do not have a band gap (Fig.
2 and 4).
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Physical Chemistry Chemical Physics Accepted Manuscript
terms [48]: kinetic (diagonal) term, attractive direct and the repulsive exchange interaction matrix
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The electronic band structures of Li2MoN2, Cl2MoN2 and F2MoN2 are depicted in Fig. 2. This
figure shows that surface functionalization changes drastically the band dispersion. The Cl2MoN2
Li2MoN2, the conduction band minimum (CBM) and the valence band maximum (VBM) are
located between ? and K points. The HSE (PBE) calculations give the band gap of 1.70 (1.24),
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1.97(1.54), 1.72(0.98) eV for Cl2MoN2, F2MoN2 and Li2MoN2 respectively which are in good
accordance with previous report [30]. In all the band structures, the band dispersion is larger for
the conduction bands. The band dispersion of valence band maximum (VBM) of F2MoN2 is the
largest near the K point causing the different absorption intensity at low energy region.
2. TOPOLOGICAL PROPERTIES
In this section we will consider the electronic structures of C-type configurations of Potassium
functionalized XN2 (K2MoN2 and K2WN2) and determine its topological properties. To check the
stability of these structures we have calculated their phonon dispersion. The calculated phonon
spectra of K2MoN2 and K2WN2 are depicted in Fig. 3. These spectra do not have any imaginary
frequency so these structures are dynamically stable. For K2MoN2 and K2WN2, the orbitalcharacter analysis and projected partial densities of states show that around Fermi level the Mo-d,
N-p and K-s orbitals are dominant. The electronic band structure of K2MoN2 (K2WN2) has been
calculated and depicted in Fig. 4 and 5. There are two main features in the band structure near to
Fermi level. The first one is a hole-like valence band at the K point and the other one is around ?
point where the valence band crosses the Fermi level and the conduction bands make a nodal ring
around the ? point as well as a Dirac cone with a small gap along ?-K. As it is clear from these
figure, there are three bands that cross each other near the ? point, the two electron-like
conduction bands from N-pz/K-s orbitals and a hole-like valence band from Mo-dz2 (W-dz2)
orbitals. The stacking sequence of Mo (W), N, K atomic planes within the unit cell is K-NMo(W)-N-K which does not preserve the space inversion symmetry. In this lattice, the Mo(W)
atomic plane is a mirror plane of structure under the mirror operation Rz that changes z to ?z.
Therefore, the structure is reflection-symmetric with respect to the Mo(W) atomic plane, which
protects the topological nodal line [8, 23]. Because of this in-plane mirror symmetry, the 2D
structure has Cs point group which only has one-dimensional irreducible representations and
. In the absence of SOC, the valence band constructed X-dz2 and the conduction band
constructed from N-pz belong to different representations of the space group, and . So the
intersection of two bands is protected by mirror reflection symmetry, forming a spinless nodal
ring on the mirror plane Kz=0. On the contrary, the valence band originated from X-dz2 and
conduction band originating from K-s have the same representation of the space group. Therefore,
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Physical Chemistry Chemical Physics Accepted Manuscript
has ? (valence band minimum) to K (conduction band minimum) indirect gap. In F2MoN2 and
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their band-crossing is gaped out causing a Dirac cone with a finite gap at a crossing point which is
0.17 eV above the Fermi level.
and valence bands consist mostly of Mo-dz2 and N-pz. This figure shows that Mo-dz2 evolves into
conduction bands and becomes prominent around 1 eV. Moreover, K-s increases below the Fermi
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level and has peak above -1 eV. This distribution of densities of states is consistent with the
orbital character analysis. Fig. 7 shows the band structure of K2MoN2 including spin-orbit
coupling. Because of the lack of space inversion symmetry in the system, in the presence of SOC
each band splits in two spin-polarized branches with opposite mirror reflection eigenvalues and
spin directions. The band splitting is more pronounced around K and increases from ? to K
causing the appearance of the gap at the band-crossing point. Although spin orbit coupling
induces tiny gap at crossing points, it gives rise a new nodal ring at crossing point along ?-K.
Such an SOC induced nodal ring have been reported in PbTaSe2 [2]. This crossing of bands with
opposite mirror reflection eigenvalues is protected by crystalline symmetry.
As mentioned before, the nodal rings are under the protection of the reflection symmetry. This
reflection symmetry can be break by moving one of the atoms slightly along the z-direction
(perpendicular to the nanosheet). In mirror-reflection systems, a band crossing forms a stable line
node when it lies on a mirror-reflection plane and two energy bands have different mirror
reflection eigenvalues. Fig. 8 shows that by breaking the reflection symmetry, two crossing bands
around ? are found to belong to the same representation of the reduced space group and the bandcrossing is gaped out. This result confirms that the nodal line ring is protected by mirror reflection
symmetry.
The existence of the nodal-line structure around ? can be proved from the effective K.p
Hamiltonian. In other words, in order to determine the accuracy of DFT results, we use the K.p
model near band inversion point. The most general form of two crossing bands around the ?
point can be written as:
? (8 ) = ?;<= 9 (8):
(3)
Where 9 (8 ) are real functions and K is relative to the ? point. := is the identity matrix and :>,/,;
are Pauli matrices for the space constructed from two bands around the ? point. The symmetries at
the ? point are mirror symmetry Rz: ? ? and time-reversal symmetry.
The time-reversal symmetry is represented by ? = :/ k where k denotes the complex conjugation
operator and :/ is the Pauli matrix. In the absence of SOC, the time-reversal symmetry is simply
given by the identity matrix times 4 which acts on the Hamiltonian as:
?? (8 )? -> = ? (?8)
7
(4)
Physical Chemistry Chemical Physics Accepted Manuscript
The calculated partial densities of states in Fig. 6 show that near to the Fermi level the conduction
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Which leads to the result that 9=,>,; (8) are even and 9/ (8) is an odd function of K. The mirror
symmetry AB = C:B , can be represented by AB = :B in the absence of SOC. The mirror reflection
AB ? D8E , 8F , 8B GAB-> = ? (8E , 8F , ?8B )
(5)
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Which requires that
9>,/ D8E , 8F , 8B G = ?9>,/ (8E , 8F , ?8B )
(6)
9; D8E , 8F , 8B G = 9; (8E , 8F , ?8B )
(7)
The constraints on 9 (8) from time-reversal and mirror reflection symmetry lead to 9> (8 ) = 0,
9/ (8) is an odd function of K and 9=,; (8) are even functions of K. So we can write
9= = I= + ?B<E I 8/
(8)
9; = = + ?B<E 8/
(9)
9/ = ?B<E K 8
(10)
The I ,K and values can be obtained by fitting to the DFT band structure. We note that the
sign of these parameters play crucial role in the existence of nodal ring [52].
The eigenvalues of Eq. 1 show that the necessary conditions for the band crossing are 9/ (8) = 0
and 9; (8 ) = 0. On the plane Kz=0, equations 4 and 5 dictate that 9>,/ (8 ) = 0 and from Eq. 7, we
have = + > 8E/ + / 8F/ = 0.The inverted band around ? leads to = > 0 and >,/ < 0. So on the
plane Kz=0, 9; is an equation for an ellipse which surrounds the ? point and leads to the bands
crossing points.
3. OPTICAL PROPERTIES
As the surface functionalized MoN2 and WN2 have similar band structures, we will only consider
the optical properties of surface functionalized MoN2 sheets. The calculated imaginary parts of the
dielectric function versus the photon energy for functionalized A2MoN2 (A=Cl, F, Li) sheets are
shown in Fi. 9. These components have been calculated within RPA approximation of Eq. (2). In
general comparison, below 3 eV, the Im ? of Li2MoN2 decreases as function of photon energy
while it increases in Cl2MoN2 and F2MoN2. In this region for both Cl2MoN2 and F2MoN2 the Im ?
has three main peaks where the first one occurs nearly at 2.2 eV in both cases. For Li2MoN2, the
Im ? starts at 2.1 eV and has the largest weight of absorption peak between 3 and 4 eV. Excluding
the large peak of Im ? of F2MoN2 above 5.5 eV, the oscillator strengths of the Im ? of Cl2MoN2
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Physical Chemistry Chemical Physics Accepted Manuscript
symmetry requires that
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and F2MoN2 are at the same level. In other words, for Cl2MoN2 and F2MoN2, the Im ? changes
nearly in the same way.
Electron-hole interactions increase the static dielectric constants and move the spectra to lower
energy (red shift) causing the fast reduction of real part of dielectric functions. The real part of
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dielectric function has negative values from 4.14 up to 4.96 eV for F2MoN2 while it has negative
values around 3.5, 5.5 and above 6.98 (up to 9 eV) for Li2MoN. The real part of dielectric
function of Cl2MoN2 does not have negative value and become negligible above 3 eV. Above 1.5
eV, Re ? of Li2MoN2 has several peaks with noticeable intensity which are distributed from 1.5 to
5 eV. It has negative values around 3.5 eV which corresponds to a high reflectivity region. In
comparison to Li2MoN2 and Cl2MoN2, the effects of electron-hole interactions on the Re ? are
more pronounced in F2MoN2 where the static dielectric constant becomes nearly double. Fig. 10
shows that the static dielectric constants are 6.98, 7.47 and 7.99 for Li2MoN2, Cl2MoN2 and
F2MoN2 respectively.
As mentioned before, the inclusion of electron-hole (e-h) interactions in the calculations of optical
absorption of functionalized MoN2 sheets reveal the presence of strongly bound excitons below
the absorption onset. Fig. 9 shows that the absorption of functionalized MoN2 including e-h
interactions and as it is clear from this figure, e-h interactions shift the absorption onsets to lower
energy (red shift) and increases the weight of main absorption peak of Im ?. In low energy part,
the Im ? of F2MoN2 has three distinct peaks where the main peak occurs at 1.3 eV. For Cl2MoN2,
two first peaks occur at the shoulder of main peak at 1.5 eV. The Im ? of Li2MoN2 has three main
parts with the prominent peaks at 1.3, 3.3 and around 5 eV so Li2MoN2 can absorb a wide range
of photon energy. For Cl2MoN2, from 2 to 8 eV, the Im ? is a monotonically decreasing function
of photon energy while for F2MoN2 and Li2MoN2 there are noticeable peaks in this region. Below
2 eV, the Im ? of F2MoN2 has the largest energy range of absorption which starts at 0.61 eV. This
energy range becomes narrower by replacing F with Cl and Li. The oscillator strengths for optical
transitions in Fig. 11 shows that for F2MoN2 the Im ? displays two strongly bound excitons below
1eV with binding energy of 1.35 and 1.08 eV respectively. Interestingly the binding energy of
first exciton is larger than corresponding one in MoS2[32]. The excitation energies of other
excitons are distributed from 1 to 2 eV suggesting possible optical applications in infrared (IR) to
red regime. In comparison to Li2MoN2, Cl2MoN2 and F2MoN2 have more excitons with lower
intensity.
The Cl2MoN2 monolayer displays the presence of several excitons with excitation energies
ranging from 1 to 2 eV where the binding energy of first exciton is 0.67 eV. For Li2MoN2, the
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Physical Chemistry Chemical Physics Accepted Manuscript
Figure 10 shows the real part of the dielectric function of surface functionalized MoN2 sheets.
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number of excitons and their binding energy are reduced. In other words, similar to molybdenum
dichalcogenides [33] there are two bound exciton below below 2 eV.
characterized by the presence of two low-energy excitonic peaks [33], the number of bound
exciton in functionalized MoN2 depend strongly on the terminated surface group. Indeed, by
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replacing the surface terminated group F with Cl or Li, the exciton numbers are reduced and
displaced to higher energies. Interestingly, the surface terminated groups strongly change the
energy distribution range of exciton which can be used to tune the optical absorption at IR and
visible light. Furthermore, as seen in Fig. 11, Li2MoN2, Cl2MoN2 and F2MoN2 show substantially
large absorption at visible light and ultraviolet light which guarantee high efficiency in the in the
utilization of solar energy.
Finally, we consider the optical properties of Na2MoN2 and K2MoN2 which have zero band gap.
Our calculations show that the electron-hole interactions do not change dramatically the
absorption spectrum and no bound exciton is observed in the absorption spectrum of Na2MoN2
and K2MoN2.
These results are in good accordance with recent first principles calculations on the optical
properties of semimetals [53, 54]. Fig. 12 shows the real and imaginary part of dielectric
functions of Na2MoN2 and K2MoN2. In both cases, the main part of absorption is below 1 eV. In
fact, the imaginary part of dielectric function decreases very fast and its main peaks are located at
low energy region. For Na2MoN2 and K2MoN2, the Im ? shows a main peak around 0.4 eV while
it has the lower oscillator strengths in the latter. Real part of dielectric function of Na2MoN2 (Fig.
12) has negative value from 0.49 eV up to 1.03 eV while for K2MoN2, the Re ? is positive at low
energy region and has negligible values above 7 eV. The static dielectric constants are 29.08 and
21.44 for Na2MoN2 and K2MoN2 respectively.
IV.SUMMARY AND CONCLUSIONS
In this work, based on density-functional theory (DFT) we have investigated the structural
properties of K2XN2 (Mo, W) nanosheets as a new topological nodal-line semimetals. Then, to
study the quasiparticle energy and optical excitations, we apply the first principle Bethe-Salpeter
equation (BSE) approach. Our calculations show that the band inversion happens between X-dz2
(x=Mo, W) and N-p/K-s orbitals which stems from crystal field effect in the nanosheet. Spin-orbit
coupling lifts the band-crossing and creates a new nodal ring above the Fermi level. The nodal
ring is under the protection of the reflection symmetry. This reflection symmetry can be broken
by moving one of the atoms slightly along z-direction (perpendicular to the nanosheet). The
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Physical Chemistry Chemical Physics Accepted Manuscript
In comparison to absorbance spectra of transition metal dichalcogenide monolayers which are
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absorption of functionalized MoN2 including e-h interactions displays the redshift of absorption
onsets causing the appearance of the strongly bound excitons below the energy gap. Electron-hole
Li2MoN2, Cl2MoN2 and F2MoN2 respectively. In comparison to Li2MoN2, Cl2MoN2 and F2MoN2
have more exciton with lower intensity. Interestingly the number of bound exciton in
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functionalized MoN2 depend strongly on the terminated surface group. By replacing the surface
terminated group F with Cl or Li, the exciton numbers are reduced respectively.
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Physical Chemistry Chemical Physics
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Fig. 1. Side view of the crystal structure of K2MoN2. The possible sites for surface termination are A (on
top of the N site), B (on top of Mo site), and C (on top of hollow site). The mirror reflection plane is shown
by Mz.
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Physical Chemistry Chemical Physics Accepted Manuscript
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DOI: 10.1039/C7CP05844F
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Fig. 2. The band structure of Cl2MoN2, F2MoN2, Na2MoN2 and Li2MoN2
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Physical Chemistry Chemical Physics Accepted Manuscript
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DOI: 10.1039/C7CP05844F
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Fig. 3. The phonon dispersions of optimized K2MoN2 and K2WN2 nanosheets.
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DOI: 10.1039/C7CP05844F
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Fig. 4. The band structure of K2MoN2 calculated without spin-orbit coupling (SOC). The insets show the
enlarged plot at band-crossing points.
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DOI: 10.1039/C7CP05844F
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Fig. 5. The orbital-projected band structures of K2XN2 (X=Mo, W) into X-dz2, N-pz and K-s.
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DOI: 10.1039/C7CP05844F
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Fig. 6. Calculated partial densities of states of K2MoN2. The enlarged plot around the Fermi level is also
shown.
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Physical Chemistry Chemical Physics Accepted Manuscript
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DOI: 10.1039/C7CP05844F
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Fig. 7. The band structure of K2MoN2 calculated with spin-orbit coupling (SOC). The enlarged plot around
band-crossing points is also shown (right).
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DOI: 10.1039/C7CP05844F
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Fig. 8. The band structure of K2MoN2 when the reflection symmetry is broken by moving the N atom
slightly along z-direction.
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DOI: 10.1039/C7CP05844F
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Fig. 9. The calculated imaginary parts of the dielectric function as a function of photon energy
for functionalized A2MoN2 (A=Cl, F, Li) sheets in RPA (solid line) and BSE (dashed line).
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DOI: 10.1039/C7CP05844F
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Fig. 10. The calculated real parts of the dielectric function as a function of photon energy
for functionalized A2MoN2 (A=Cl, F, Li) sheets in RPA (solid line) and BSE (dashed line).
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Physical Chemistry Chemical Physics Accepted Manuscript
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DOI: 10.1039/C7CP05844F
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Fig. 11. The calculated imaginary parts of the dielectric function as a function of photon energy for
functionalized A2MoN2 (A=Cl, F, Li) sheets in BSE. Vertical (blue) bars represent the relative oscillator
strengths for the optical transitions. Red dashed lines indicate the HSE band gap.
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DOI: 10.1039/C7CP05844F
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Fig. 12. The calculated real (dashed) and imaginary (solid line) parts of the dielectric function as a
function of photon energy for functionalized A2MoN2 (A=Na, K) sheets in RPA.
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DOI: 10.1039/C7CP05844F
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