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Solid State Communications

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Communication

Optimized tight binding parameters for single layer honeycomb borophene

Liyan Zhua,b, Tingting Zhanga,b,∗

a Department of Physics, Jiangsu Key Laboratory for Chemistry of Low-Dimensional Materials, Huaiyin Normal University, Huai'an, People's Republic of Chinab Jiangsu Key Laboratory of Modern Measurement Technology and Intelligent Systems, Huaiyin Normal University, Huai'an, People's Republic of China

A R T I C L E I N F O

Communicated by Ralph Gebauer

Keywords:BoropheneTight bindingElectronic band structureDensity of states

A B S T R A C T

Motivated by experimental realization of honeycomb borophene (HB) on an Al(111) surface, we propose a non-orthogonal Slate-Koster tight binding (TB) model for single layer HB, of which parameters are optimized via acombination of the genetic algorithm and the simplex method using electronic structures obtained from densityfunctional theory (DFT) as targets. Our proposed Slate-Koster parameters can well produce both energies andorbital compositions of electronic bands of a single layer HB. The direct calculation of band structure and atomicorbital resolved density of states show excellent agreement with DFT results. The highly accurate TB modelwould facilitate the future large-scale atomistic modeling on electronic, optical, and transport properties of HBbased nano-materials.

1. Introduction

Since the successful exfoliation of graphene [1], exploration of two-dimensional (2D) materials has received extensive interest [2]. As a leftneighbor of carbon, great effort has been devoted to searching 2D al-lotropes of boron [3], i.e. borophene. Extensive studies on small sizedboron clusters suggest that the triangular motif is a building block forconstructing large sized boron clusters [4,5]. Inspired by these findings,a theoretical study proposed that the buckled triangular structure ofborophene could be a stable allotrope of 2D boron [6]. However, thefurther investigation revealed that the cohesive energy of borophenecan be remarkably enhanced by removing a few fractions of boronatoms to create hexagonal holes into the pristine triangular lattice [7].For example, Tang et al. [7] proposed two energetically favorable 2Dallotropes of boron with a porosity of 1∕9 and 1∕7, which are coined asα- and β-borophene. Currently, advanced algorithms, e.g., particleswarm optimization [8], cluster expansion [9], genetic algorithm [10],have been employed to search candidates of 2D boron. Besides theadvance in theoretical studies, experimental realizations of 2D boronhave been achieved on Ag(111) by Manix et al. [11] and Feng et al.[12], in which the rectangular and rhombohedral phases can match theallotropes of borophene with a hexagonal hole density of 1∕6 and 1∕5,respectively [13].

In addition to diversity in 2D allotropes, borophene also possessesmany outstanding physical and chemical properties [13]. For example,Zhang and co-workers predicted a high and anisotropic in-plane

modulus for borophene ranging from 189 to 399 N/m [14], which iscomparable to that of graphene [15] and hexagonal boron nitride [16].Besides superior mechanical strength, structure diversity renders many2D allotropes with nontrivial electronic structures and emergent fer-mions, e.g., Dirac [17] and triplet [18] fermions. Furthermore, bor-ophene was also predicted to be the first known materials with high-frequency plasmons in the visible regions [19]. Such intriguing prop-erties make borophene a promising candidate for many applications,e.g., electronic and optoelectronic devices. On the other hand, the in-triguing electronic properties of graphene [20], massless Dirac fermionsresulted from a linear energy-momentum relation, has aroused tre-mendous interest in exploring honeycomb alternatives to graphene. So,it is natural to ask whether the honeycomb borophene (HB) can be astable 2D allotrope of boron. Unfortunately, it turns out that the free-standing HB is kinetically unstable. This is because each boron atom hasonly three valence electrons. However, a boron atom can form three σbonds with three nearest neighbors as well as one π bond. So, all threevalence electrons will prefer to occupy the bonding σ orbitals; while thebonding π orbital is almost unoccupied due to electron deficiency ofboron [7,21,22]. The lacking of bonding π orbital is detrimental to thestability of HB.

However, the experimentally realized borophene is not the α-bor-ophene [11,12], the lowest energy structure predicted by theoreticalsimulations [7]. It might be due to the charge doping from Ag(111) tothe borophene. The study carried out by Zhang et al. [23] clearly de-monstrated that the lowest energy allotropes of borophene significantly

https://doi.org/10.1016/j.ssc.2018.08.003Received 24 June 2018; Received in revised form 1 August 2018; Accepted 4 August 2018

∗ Corresponding author. Department of Physics, Jiangsu Key Laboratory for Chemistry of Low-Dimensional Materials, Huaiyin Normal University, Huai'an, People'sRepublic of China.

E-mail address: ttzhang@hytc.edu.cn (T. Zhang).

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Available online 06 August 20180038-1098/ © 2018 Elsevier Ltd. All rights reserved.

T

depends on the charge doped to the borophene. For example, the op-timal porosity increases with the amount of charge doped into bor-ophene [24]. Most interestingly, if the doped charge is greater than 0.5electrons per boron atom, the HB would become the lowest energystructure of 2D boron [24]. This kind of recipe has been realized by Wuet al. [25] to synthesize the HB on an Al(111) substrate using the mo-lecular beam epitaxy method. For borophene supported on Al(111),there are around 0.7 electrons transferred from Al(111) to each boronatom according to their calculations [25]. Thus, the realization of HBon Al(111) is highly expected.

Motivated by the successful synthesis of HB, we aim to propose aprecise tight binding (TB) model for the single layer HB. Through acombination of genetic algorithm [26] and the simplex method, a set ofoptimized Slate-Koster parameters has been obtained, which could wellreproduce the band structure and density of states (DOS) of HB ascompared with the first principle results. The accurate TB model mightfacilitate future numerical studies on electronic and optical propertiesof HB on a large scale.

2. Crystal structure and TB model

2.1. Crystal structure

In a honeycomb phase of borophene, there are two atoms in a pri-mitive cell as labeled as A and B in Fig. 1. The two Bravais primitivelattice vectors are

=a a( , 0, 0)T1 (1)

= −( )a a a, , 0T

212

32 (2)

where a is the lattice constant of HB, namely, 2.92 Å.In a honeycomb lattice, there are 3 nearest neighbors (NNs) of a

central atom, and 6(3) next (third) NNs. The vectors connecting nearestneighbors (NN) of a A- or B-typed central atom, δ1, are

= ±± ( )δ a0, , 0T

1,1 1

3 (3)

= ± −± ( )δ a a,T

2,1 1

21

2 3 (4)

= ± − −± ( )δ a a, , 0 ,T

3,1 1

21

2 3 (5)

where the sign + (−) corresponds to the central atom being of an A(B)type. In a similar manner, the vectors connecting second NNs are givenby,

=δ a( , 0, 0) ,T12 (6)

= −δ a( , 0, 0) ,T22 (7)

= ( )δ a a, , 0 ,T

32 1

23

2 (8)

= −( )δ a a v, , 0 ,42 1

23

2 (9)

= −( )δ a a, , 0 ,T

52 1

23

2 (10)

= − −( )δ a a, , 0 ;T

62 1

23

2 (11)

and the third NNs for A- and B-typed central atoms are

= ± −± ( )δ a0, , 0 ,T

1,3 2

3 (12)

= ±± ( )δ a a, , 0 ,T

2,3 1

3 (13)

= ± −± ( )δ a a, , 0 .T

3,3 2

3 (14)

2.2. TB model

The basis for boron (ψ) is taken to be four cubic harmonic orbitals,namely, 1 s orbital plus three p orbitals (px, py, pz) for each boron atom.Since the primitive cell of HB contains two boron atoms, theHamiltonian is thus eight dimensional with the following basis for bothboron atoms A and B in a primitive cell,

∣ ⟩ = ∣ ⟩ ∣ ⟩ ∣ ⟩ ∣ ⟩ ∣ ⟩ ∣ ⟩ ∣ ⟩ ∣ ⟩( )φ φ φ φ φ φ φ φ φ, , , , , , ,sA

pA

pA

pA

sB

pB

pB

pB T

x y z x y z (15)

In order to fulfill Bloch's theorem, the Bloch basis function, ∣ϕiα(k)⟩,can be constructed as a superposition of atomic basis functions,

∑∣ ⟩ = ∣ ⟩⋅ +ϕ e φk( ) ,iαi r

αi

R

k R( )i

(16)

where k is a crystal momentum, and i represents the boron atoms in aprimitive cell, namely, A and B as shown in Fig. 1; while R denotes oneof lattice vectors connecting primitive cells. Hence, each Bloch eigen-state can be expanded as a linear combination of such Bloch basisfunctions,

∑∣ ⟩ = ∣ ⟩c ϕk kΨ ( ) ( )niα

iαn

iαk

(17)

Then, the secular equation, H∣Ψ⟩= E∣Ψ⟩, can be transformed into amatrix form,

⋅ = ⋅EH k C O C( ) n n nk k k (18)

where

= ∑ ⟨ ⟩⋅ δH e φ H φ(0)| | ( ) ,δiα jβ l t

iαi

βj

tk

, ,tl

(19)

= ∑ ⟨ ∣ ⟩⋅ δO e φ φ(0) ( ) .δiα jβ l t

iαi

βj

tk

, ,tl

(20)

These two terms are hopping and overlap matrix elements betweenthe α orbital of atom i and the β orbital of atom j, respectively. Thehopping matrix element between the different orbitals can be calcu-lated in terms of Slater–Koster (SK) parameters (see Table 1) [27], Vssσ

l ,Vspσ

l , Vppσl , Vppπ

l , where l= 1, 2, and 3, representing the first, second,and third NNs, respectively. Besides the hopping matrix elements, theon-site energies for s and p orbitals are denoted by Es and Ep.

Correspondingly, the overlap matrix elements between the differentatomic orbitals can also be calculated using SK parameters Ossσ

l , Ospσl ,

Oppσl , Oppπ

l in a similar manner. Therefore, there will be 26 parametersneeded to be fitted from first principle calculations.

Fig. 1. Atomic structure of HB (left) and the first Brillouin zone of HB (right).The red rombohedra is the primitive cell of single layer HB; while the red, blue,and green arrows denote the first, second, and third nearest neighbors of a B-typed central atom. (For interpretation of the references to colour in this figurelegend, the reader is referred to the Web version of this article.)

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3. Density functional calculations and optimization of SKparameters

All first principle calculations were carried out using SIESTA [28]within the framework of density functional theory (DFT). The generalgradient approximation to exchange-correlation functional, para-meterized by Perdew, Burke, and Ernzerhof [29], was used to describethe interaction between electrons; while the interaction between elec-trons and ions was modeled by norm-conserving pseudopotential.[30]The wavefunctions were expanded by numerical basis sets, namely,double zeta basis plus polarization orbitals (DZP). The energy cutoffwas set to be 200 Ry. The Brillouin zone was sampled by a k-mesh of25×25×1. The structure of HB was fully relaxed with a conjugategradient method until the force acting on each atom was less than1.0×10−2 eV/Å.

Fig. 2 shows the band structure of HB obtained from DFT calcula-tions which is overall similar to that of graphene. Each boron atomforms three σ bonds to three NNs through sp2 hybridization of the s, px,and py orbitals. The σ bonds are highly localized between boron atoms,and there is a large energy gap separating the bonding and anti-bondingσ states. While the coupling of the remaining pz orbitals results in de-localized π bonds. However, the bonding and anti-bonding π statestouch at six points in the Brillouin zone, which is coined as Dirac points.The cone shape of band structure gives rise to many fascinating prop-erties in graphene. Thus, the searching of band structure with linearcrossing points becomes an interesting topic during the last decades.But the number of valence electrons is three for each boron atom, whichis less than the available bonding orbitals, namely, 4. So most of valenceelectrons occupy the three σ bonding orbitals; while the π bondingstates are mostly unoccupied. Thus, the Dirac point in HB is around3.5 eV higher than the Fermi level.

We next turn to fit SK Parameters for HB, which is normally a dif-ficult job. One must correctly reproduce the position of energy bands aswell as their orbital compositions. To obtain accurate SK parameters,we employ a two-stage fitting strategy. Firstly, we adopt the geneticalgorithm to optimize the SK parameters [26]. Initially, a population of200 individuals was created randomly, and then evaluated fitnessfunction for the population. The fitness function is defined below,

∑= −F w E E( ) ,n

TBn

DFTn

kk

k k

,

2

(21)

where wk is weight at the k-point of k. Through selection, crossover,and mutation operations, we can roughly obtain some parameter setswhich reproduce the band structures in good agreement with firstprinciple results. Then, we further refine these parameter sets using theNelder-Mead simplex algorithm [34].

The best-fitted parameters are listed in Table 2, and correspondingband structure of HB reproduced from this set of parameters is plottedin Fig. 2. Our proposed SK parameters can well reproduce the bandstructure of HB as compared with the DFT results. To further examinethe orbital components, we also plotted the atomic orbital resolveddensity of states in Fig. 3, and compare them with the DFT results.Apparently, the DOS obtained from our TB model can well match thatfrom DFT calculations, including both shapes and positions of van Hovesingularities. As seen from Fig. 3, the DOS of pz states vanishes atE=3.5 eV, which separates the bonding and anti-bonding π-orbitals;while in the plot of band structure, the bonding and anti-bonding π-orbitals exactly touch at six vertexes of Brillouin zone, i.e., the K and K′points, which is intuitively demonstrated in the three-dimensional plotof π-bands (see Fig. 4). However, the weakness of HB, compared withgraphene, is that the Dirac point is around 3.5 eV higher than the Fermilevel. So, it is hard to observe the excitation of massless Dirac fermionsin freestanding HB. One may need to identify suited insulating sub-strates which could not only dope sufficient electrons to the HB and alsohave little overlap with the π-orbitals of HB. The sufficient chargedoping could eventually shift the Dirac point close to the Fermi level.

In order to further validate our TB model in describing opticalproperties of HB, we calculate the imaginary part of dielectric functionscontributed by the interband transition according to the formula givenin Refs. [35–37]. The corresponding results obtained by DFT is alsoshown in Fig. 5 for comparison. Overall, the two methods give rise tosimilar results, which confirms the high accuracy of our TB model. Themajor peak (∼0.8 eV) is due to transitions occurred close to the middlepoint of Γ−K path, which is indicated as the blue shaded region inFig. 2. In such a region, two bands have a slope similar to each other,thus leading to a large joint DOS. It should be noted here that the di-electric functions we obtained are far from accurate, in which we onlyconsider the contribution from interband transitions. Free carrier con-tribution to the dielectric function is also very important to metallicsystems. Sophistic methods have to be used in order to get accuratedielectric functions.

For the purpose of comparison, we also list the optimized SKparameters of 2D graphene [31,32] and 1D graphene nanoribbon [33]in Table 2. Apparently, the overlapping interaction between out-of-plane pz orbitals of nearest neighboring atoms (Vppπ) in HB is smallerthan that in graphene or graphene nanoribbon, which can be attributed

Table 1Calculation of hopping matrix elements, ⟨ ⟩φ H φ| |α

iβj , using SK parameters [27]. The symbol l, m, n denote the direction cosines of the vector connecting atom i to

neighbor atom j.

A s px py pz

s Vssσ lVssσ mVssσ nVssσpx −lVssσ l2Vppσ + (1 − l2)Vppπ lm(Vppσ− Vppπ) ln(Vppσ− Vppπ)py −mVssσ lm(Vppσ− Vppπ) m2Vppσ + (1 − m2)Vppπ mn(Vppσ−Vppπ)pz −nVssσ ln(Vppσ− Vppπ) mn(Vppσ− Vppπ) n2Vppσ + (1 − n2)Vppπ

Fig. 2. Comparison band structure obtained from DFT calculations (black) andour proposed TB model (red). The blue-shaded area indicates where stronginterband optical transitions occur. (For interpretation of the references tocolour in this figure legend, the reader is referred to the Web version of thisarticle.)

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to the long BeB bond length, 1.68 Å, which is around 18% longer thanthat of CeC bonds. The small Vppπ in HB would result in a suppressedFermi velocity around the Dirac cone, i.e. 0.81×106m/s, as comparedwith that in graphene (∼1×106m/s [20]) since the Fermi velocity isproportional to the magnitude of Vppπ [20].

Before concluding, a few remarks should be addressed. Firstly, ourTB model is only suited to pristine single layer HB. It is not able to studythe interaction of single layer of HB with other atoms, e.g. functiona-lization of HB with hydrogen and halogen atoms and adatoms of extra Batoms onto HB. It is because the SK parameters are sensitive to thebonding environment, e.g., bond lengths and bond angles. In order tomake the TB model transferable, one has to consider the bond-length

dependence of SK parameters. In addition, our TB model does not in-clude layer-layer interaction yet. It is partially because only single layerHB has been experimentally realized so far. If bilayer HB supported onAl(111), the electrons doped into the top layer of HB is almost zeroaccording to our DFT calculations. As already pointed out by Zhanget al. [24], only if sufficient electrons are doped into HB (i.e. ~ 0.5electrons per boron atoms), can the honeycomb allotrope of 2D boronbe the lowest energy configuration. Hence, the bilayer or multilayer HBmay not be easily synthesized as the single layer counterpart.

4. Conclusion

In summary, we have optimized the SK parameters for a single layerHB up to the third NNs. This optimized SK parameter set can reproducea highly accurate band structure and DOS of HB. So, it could serve as asound starting point for further large-scale numerical studies on elec-tronic and optical properties of single layer HB in future. However, theDirac point still has an energy higher than the Fermi level. Thus, itwould demand to find out a suited dielectric substrate on which thesufficient charge doping may not only stabilize the HB, but also shift theDirac point close to the Fermi level. Then, it would be promising toutilize the HB to fabricate optoelectronic devices.

Acknowledgments

Authors acknowledge support from National Science Foundation ofChina (NSFC) (Grant No. 11504122 and 11704141), Natural Science

Table 2Optimized SK parameters (in eV) for a single layer HB.

Parameter V1 V2 V3 O1 O2 O3

ssσ −2.796 3.687×10−2 −7.119×10−2 −7.783×10−3 −3.873× 10−2 6.655× 10−2

spσ 3.754 5.543×10−2 −8.432×10−3 −1.963×10−1 3.377× 10−2 −1.375× 10−2

ppσ 2.866 1.890×10−1 6.916× 10−1 −2.517×10−1 −7.771× 10−3 −1.280× 10−2

ppπ −2.213 9.670×10−2 −3.036×10−3 −1.339×10−2 2.136× 10−3 5.992× 10−2

Es −1.030Ep 3.673

ppπ(G) [31] −2.97 −0.073 −0.33 0.073 0.018 0.026ppπ(G) [32] −2.78 −0.15 −0.095 0.117 0.004 0.002ppπ(GNR) [33] −2.756 −0.071 −0.38 0.093 0.079 0.070

Fig. 3. Atomic orbital resolved DOS obtained from DFT calculations (top) andour proposed TB model (bottom).

Fig. 4. Three-dimensional plot of bonding and anti-bonding π bands.

Fig. 5. Imaginary dielectric function versus photon energy obtained from DFTcalculations and our proposed TB model.

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Foundation of the Higher Education Institutions of Jiangsu Province(Grant No. 15KJB140001).

References

[1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos,I.V. Grigorieva, A.A. Firsov, Electric field effect in atomically thin carbon films,Science 306 (5696) (2004) 666–669.

[2] A.K. Geim, K.S. Novoselov, The rise of graphene, Nat. Mater. 6 (3) (2007) 183.[3] Z. Zhang, E.S. Penev, B.I. Yakobson, Two-dimensional boron: structures, properties

and applications, Chem. Soc. Rev. 46 (22) (2017) 6746–6763.[4] I. Boustani, Systematic ab initio investigation of bare boron clusters:mdetermina-

tion of the geometryand electronic structures of bn (n=2–14), Phys. Rev. B 55(1997) 16426–16438.

[5] H.-J. Zhai, B. Kiran, J. Li, L.-S. Wang, Hydrocarbon analogues of boron clus-ters—planarity, aromaticity and antiaromaticity, Nat. Mater. 2 (12) (2003) 827.

[6] M.H. Evans, J.D. Joannopoulos, S.T. Pantelides, Electronic and mechanical prop-erties of planar and tubular boron structures, Phys. Rev. B 72 (2005) 045434.

[7] H. Tang, S. Ismail-Beigi, Novel precursors for boron nanotubes: the competition oftwo-center and three-center bonding in boron sheets, Phys. Rev. Lett. 99 (11)(2007) 115501.

[8] X. Wu, J. Dai, Y. Zhao, Z. Zhuo, J. Yang, X.C. Zeng, Two-dimensional boronmonolayer sheets, ACS Nano 6 (8) (2012) 7443–7453.

[9] E.S. Penev, S. Bhowmick, A. Sadrzadeh, B.I. Yakobson, Polymorphism of two-di-mensional boron, Nano Lett. 12 (5) (2012) 2441–2445.

[10] X.-F. Zhou, X. Dong, A.R. Oganov, Q. Zhu, Y. Tian, H.-T. Wang, Semimetallic two-dimensional boron allotrope with massless Dirac fermions, Phys. Rev. Lett. 112 (8)(2014) 085502.

[11] A.J. Mannix, X.-F. Zhou, B. Kiraly, J.D. Wood, D. Alducin, B.D. Myers, X. Liu,B.L. Fisher, U. Santiago, J.R. Guest, et al., Synthesis of borophenes: anisotropic,two-dimensional boron polymorphs, Science 350 (6267) (2015) 1513–1516.

[12] B. Feng, J. Zhang, Q. Zhong, W. Li, S. Li, H. Li, P. Cheng, S. Meng, L. Chen, K. Wu,Experimental realization of two-dimensional boron sheets, Nat. Chem. 8 (6) (2016)563.

[13] A. Mannix, Z. Zhang, N. Guisinger, B. Yakobson, M. Hersam, Borophene as a pro-totype for synthetic 2d materials development, Nat. Nanotechnol. 13 (6) (2018)444–450.

[14] Z. Zhang, Y. Yang, E.S. Penev, B.I. Yakobson, Elasticity, flexibility, and idealstrength of borophenes, Adv. Funct. Mater. 27 (9) (2017) 1605059.

[15] C. Lee, X. Wei, J.W. Kysar, J. Hone, Measurement of the elastic properties andintrinsic strength of monolayer graphene, Science 321 (5887) (2008) 385–388.

[16] M. Topsakal, S. Cahangirov, S. Ciraci, The response of mechanical and electronicproperties of graphane to the elastic strain, Appl. Phys. Lett. 96 (9) (2010) 091912.

[17] B. Feng, O. Sugino, R.-Y. Liu, J. Zhang, R. Yukawa, M. Kawamura, T. Iimori, H. Kim,Y. Hasegawa, H. Li, et al., Dirac fermions in borophene, Phys. Rev. Lett. 118 (9)(2017) 096401.

[18] M. Ezawa, Triplet fermions and Dirac fermions in borophene, Phys. Rev. B 96 (3)

(2017) 035425.[19] Y. Huang, S.N. Shirodkar, B.I. Yakobson, Two-dimensional boron polymorphs for

visible range plasmonics: a first-principles exploration, J. Am. Chem. Soc. 139 (47)(2017) 17181–17185.

[20] A.C. Neto, F. Guinea, N.M. Peres, K.S. Novoselov, A.K. Geim, The electronic prop-erties of graphene, Rev. Mod. Phys. 81 (1) (2009) 109.

[21] X.-B. Li, S.-Y. Xie, H. Zheng, W.Q. Tian, H.-B. Sun, Boron based two-dimensionalcrystals: theoretical design, realization proposal and applications, Nanoscale 7 (45)(2015) 18863–18871.

[22] L. Kong, K. Wu, L. Chen, Recent progress on borophene: growth and structures,Front. Physiol. 13 (3) (2018) 138105.

[23] Z. Zhang, S.N. Shirodkar, Y. Yang, B.I. Yakobson, Gate-voltage control of borophenestructure formation, Angew. Chem. Int. Ed. 129 (48) (2017) 15623–15628.

[24] S.N. Shirodkar, E.S. Penev, B.I. Yakobson, Honeycomb boron: alchemy on alu-minum pan? Sci. Bull. 63 (5) (2018) 270–271.

[25] W. Li, L. Kong, C. Chen, J. Gou, S. Sheng, W. Zhang, H. Li, L. Chen, P. Cheng, K. Wu,Experimental realization of honeycomb borophene, Sci. Bull. 63 (5) (2018)282–286.

[26] G. Klimeck, R.C. Bowen, T.B. Boykin, C. Salazar-Lazaro, T.A. Cwik, A. Stoica, Sitight-binding parameters from genetic algorithm fitting, Superlattice. Microst. 27(2–3) (2000) 77–88.

[27] J.C. Slater, G.F. Koster, Simplified lcao method for the periodic potential problem,Phys. Rev. 94 (6) (1954) 1498.

[28] J.M. Soler, E. Artacho, J.D. Gale, A. García, J. Junquera, P. Ordejón, D. Sánchez-Portal, The siesta method for ab initio order-n materials simulation, J. Phys.Condens. Matter 14 (11) (2002) 2745.

[29] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation madesimple, Phys. Rev. Lett. 77 (18) (1996) 3865.

[30] D.R. Hamann, M. Schluter, C. Chiang, Norm-conserving pseudopotentials, Phys.Rev. Lett. 43 (1979) 1494–1497.

[31] S. Reich, J. Maultzsch, C. Thomsen, P. Ordejon, Tight-binding description of gra-phene, Phys. Rev. B 66 (3) (2002) 035412.

[32] R. Kundu, Tight-binding parameters for graphene, Mod. Phys. Lett. B 25 (03) (2011)163–173.

[33] V.-T. Tran, J. Saint-Martin, P. Dollfus, S. Volz, Third nearest neighbor para-meterized tight binding model for graphene nano-ribbons, AIP Adv. 7 (7) (2017)075212.

[34] J.A. Nelder, R. Mead, A simplex method for function minimization, Comput. J. 7 (4)(1965) 308–313.

[35] A. Miranda, V. Ponomaryov, L.N. de Rivera, R. Vazquez, M. Cruz-Irisson, Dielectricfunction in semi-empirical tight-binding theory applied to crystalline diamond,Mathematical Methods in Electromagnetic Theory, 2008. MMET 2008. 12thInternational Conference on, IEEE, 2008, pp. 253–255.

[36] T. Sandu, Optical matrix elements in tight-binding models with overlap, Phys. Rev.B 72 (12) (2005) 125105.

[37] L.L.Y. Voon, L. Ram-Mohan, Tight-binding representation of the optical matrixelements: theory and applications, Phys. Rev. B 47 (23) (1993) 15500.

L. Zhu, T. Zhang Solid State Communications 282 (2018) 50–54

54