This journal is©The Royal Society of Chemistry 2019 J. Mater. Chem. C

Cite this:DOI: 10.1039/c9tc05783h

Superconductivity in predicted two dimensionalXB6 (X = Ga, In)†

Luo Yan,abcde Tao Bo,f Peng-Fei Liu, ab Liujiang Zhou, e Junrong Zhang,ab

Ming-Hua Tang, cd Yong-Guang Xiao*cd and Bao-Tian Wang *abg

Recently, two-dimensional (2D) rect-AlB6 [B. Song, Y. Zhou, H.-M. Yang, J.-H. Liao, L.-M. Yang, X.-B.

Yang and E. Ganz, J. Am. Chem. Soc., 2019, 141, 3630–3640] has been reported to be an excellent system,

exhibiting intrinsic superconductivity. Here, on the basis of the CALYPSO structure prediction method and

first-principles calculations, we predict two novel systems of 2D XB6 (rect- and hex-XB6) (X = Ga, In), verify

their thermal stability and calculate their mechanical properties. Then, we study their bonding nature,

electron and phonon properties, and electron–phonon coupling (EPC). The results show that rect-,

hex-GaB6, rect- and hex-InB6 are intrinsic phonon-mediated superconductors with a superconducting

transition temperature (Tc) of 1.67, 14.02, 7.77 and 4.83 K, respectively. Furthermore, we study the

influence of strain on these intrinsic superconductors. Their Tc can be modulated to some extent,

especially for hex-InB6, the Tc of which can be enhanced to 19.23 K under the compressive strain of 2%.

According to our calculations, they may be prepared by growing on suitable substrates, such as Cu(100)

and Pt(111). Our findings may broaden the configurations of 2D superconducting materials and will

stimulate more efforts in this field.

1 Introduction

2D materials have historically been one of the most extensivelystudied classes of materials, because of the qualitative changesin their physical and chemical properties due to the quantumsize effect, which is related to their nanosized thickness.1 Sincethe exfoliation of graphene in 2004,2 2D materials, such ash-BN,3 silicene,4,5 phosphorene,6,7 borophene,8,9 MoS2,

10,11 etc.,have grown dramatically. The family of 2D materials, includingmetals, semimetals, insulators, and semiconductors with directand indirect bandgaps,12–15 can be composed by the elementsof the periodic table.16 It is always meaningful for materialscience innovation and potential technology application17,18 todesign novel 2D materials with unique atomistic configurations

and exotic properties. Recently, rapid advancements in nano-technologies19–25 have greatly promoted the synthesis of 2Dmaterials in experiments. Even so, some experiments still fail todetermine and design novel 2D structures.26

Owing to the significant progress in both computationalpower and basic materials theory, many crystal structure predictiontechnologies have been developed and applied, including simulatedannealing,27 minima hopping,28 basin hopping,29 metadynamics,30

the genetic algorithm,31–33 the random sampling method,34

particle swarm optimization,26,35 etc. Based on these technologies,many frontier fields have been stimulated, for examples,batteries,36–40 high pressure phases,41,42 ferromagnetism,43–46

thermoelectricity,47,48 and even superconductivity.49–51 Thus, itis reliable to enrich new materials and explore exotic propertiesby crystal structure prediction together with the state-of-artdensity-functional theory (DFT).

Since borophene has been experimentally synthesized,52–54

many 2D borides have attracted much attention. Qu et al. havepredicted a planar monolayered TiB4 molecule and furtherdesigned a series of other 2D transition metal borides (TMB4,TM = V, Cr, Mo, W and Os) with quasi-planar octacoordinateTM atoms.55 A new 2D Dirac material, FeB2, with a Fermi velocityapproaching graphene has been explored by Zhang et al.56 Threeplanar FeB6 monolayers (a-FeB6, b-FeB6 and g-FeB6) with uniqueelectronic and optical properties have been reported by Zhanget al.57 And then, another sandwiched tri-FeB6 with unusualnegative Poisson’s ratio properties induced by oxidization was

a Institute of High Energy Physics, Chinese Academy of Sciences (CAS),

Beijing 100049, China. E-mail: wangbt@ihep.ac.cnb Spallation Neutron Source Science Center, Dongguan 523808, Chinac Key Laboratory of Key Film Materials & Application for Equipments (Hunan

Province), School of Material Sciences and Engineering, Xiangtan University,

Xiangtan, Hunan, 411105, China. E-mail: ygxiao@xtu.edu.cnd Hunan Provincial Key Laboratory of Thin Film Materials and Devices,

School of Material Sciences and Engineering, Xiangtan University, Chinae Institute of Fundamental and Frontier Sciences, University of Electronic Science

and Technology of China, Chengdu 610054, Chinaf Songshan Lake Materials Laboratory, Dongguan, Guangdong, 523808, Chinag Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan,

Shanxi 030006, China

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9tc05783h

Received 22nd October 2019,Accepted 13th December 2019

DOI: 10.1039/c9tc05783h

rsc.li/materials-c

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studied by Li et al.58 A graphene-like WB4 molecule exhibiting adouble Dirac cone and higher charge mobility after opening thegap has also been predicted by Zhang et al.59 Besides, in ourformer works, we have also presented several 2D boride super-conductors (Mo2B2 and W2B2).

51,60 More recently, 2D rect-AlB6

with high stability, a unique motif, triple Dirac Cones, andsuperconductivity has been systematically studied by Songet al.,17 catching our interest. Its relatively low superconductingtransition temperature (Tc = 0.95 K) can be caused by the strongbonding between B and Al, which has constrained the vibrationsof B atoms. Considering this, whether the other atoms (Ga andIn) in the same column of the periodic table can improve thesuperconductivity in these systems needs to be explored.

In the present work, we predict four novel stable structures of2D XB6 (X = Ga, In) (labled as rect-, hex-GaB6, rect- and hex-InB6and presented in Fig. 1) based on the particle swarm optimizationmethod26,35 and DFT. Then, we also demonstrate that they arethermally and mechanically stable. Subsequently, through sys-tematic first-principles calculations, their bonding types, electronicproperties and even superconductivity are studied. Results showthat our obtained systems are all not only intrinsic metals, but alsointrinsic superconductors with Tc values of 1.67, 14.02, 7.77 and4.83 K, respectively. To simulate the reality that crystals areprepared on suitable substrates, we calculate their substrate-supported superconductivity under biaxial strain and also checksome suitable substrates.

2 Computational methods2.1 Technical details

By utilizing the swarm intelligence-based CALYPSO code,26,35

many new structures of 2D gallium boride and indium boride areobtained and ranked in order of enthalpy from low to high. Then,we select some different structures with relatively low energy formore accurate optimization and phonon spectrum calculations.

The structure optimization is performed at the DFT level, asimplemented in the Vienna ab initio simulation package (VASP).61

The Perdew–Burke–Ernzerhof (PBE)62 form of the generalizedgradient approximation (GGA) is chosen for the exchange–correlation functional to describe the electron interactions,and the electron–ion interactions are described by the projector-augmented-wave (PAW) potentials. After the convergence tests, aplane-wave cutoff of 450 eV is used for the kinetic energy. Theuniform G-centered 16 � 18 � 1 and 16 � 16 � 1 k-point meshesare adopted for rect- and hex-XB6, respectively. We set a vacuumthickness of 15 Å for our 2D models to avoid the nonphysicalcoupling between adjacent layers. The maximum force on theatoms in the unit cell is less than 0.005 eV Å�1. We perform Born–Oppenheimer ab initiomolecular dynamics (AIMD) simulations ata series of temperatures (300, 500, 700, 1000 and 1300 K) with theconstant number, volume, temperature (NVT) ensemble, lastingfor 10 ps with a time step of 1 fs, to assess the thermalstabilities of our predicted structures. 3 � 3 � 1 supercellsare adopted and the temperature is controlled using the Nose–Hoover thermostat.63

The QUANTUM ESPRESSO (QE) code64,65 is used to calculatethe electronic structures and superconducting properties. Afterfull convergence, the norm-conserving PBE pseudopotentials66

are chosen to describe the electron–ion interactions. The plane-wave kinetic energy cutoff and the charge-density cutoff arechosen to be 70 and 280 Ry, respectively. The Methfessel–Paxton smearing width of 0.02 Ry is used. The self-consistentelectron densities in rect- and hex-XB6 are calculated on 30 �36 � 1 and 36 � 36 � 1 k-point grids, respectively. We reoptimizeour obtained structures within QE until the Hellman-Feynmanforces on each atom are less than 10�6 Ry Bohr�1. The dynamicalmatrices and the EPCmatrix elements are calculated on 5� 6� 1and 6 � 6 � 1 q-meshes for rect- and hex-XB6, respectively. Thephonon modes and the EPC are calculated using the density-functional perturbation theory (DFPT)67 and the Eliashbergtheory.68,69

2.2 EPC formalism

The magnitude of the EPC lqn is calculated by the Migdal–Eliashberg equation,70,71

lqn ¼gqn

phNðEFÞoqn2; (1)

where gqn is the phonon linewidth, oqn is the phonon frequency,and N(EF) is the electronic density of states (DOS) at the Fermilevel. We can estimate gqn by

gqn ¼2poqn

OBZ

Xk;n;m

jgnkn;kþqmj2dðekn � eFÞdðekþqm � eFÞ; (2)

where OBZ is the volume of the Brillouin zone (BZ), ekn and ek+qmrepresent the Kohn–Sham energy, and gnkn,k+qm represents theEPC matrix element that can be determined self-consistently bythe linear response theory.72 The Eliashberg electron–phonon

Fig. 1 Top and side views for 2D (a and b) rect- and (c and d) hex-XB6

(X = Ga, In). The B and Ga(In) atoms are denoted by red and silver spheres,respectively.

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spectral function a2F(o) and the total EPC constant l can beestimated by

a2FðoÞ ¼ 1

2pNðEFÞXqn

gqnoqn

dðo� oqnÞ (3)

and

lðoÞ ¼ 2

ðo0

a2FðoÞo

do; (4)

respectively. The logarithmic average frequency olog can becalculated by

olog ¼ exp2

l

ð10

dooa2FðoÞlogo

� �: (5)

Then, one can calculate the superconducting transition tem-perature Tc by

Tc ¼olog

1:2exp � 1:04ð1þ lÞ

l� m�ð1þ 0:62lÞ

� �(6)

where an effective screened Coulomb repulsion constant m* =0.173–77 can be used.

3 Results and discussion3.1 Atom structures

Following the directions of the recent work,17 we therefore onlyexplored the possible 2D ‘‘Ga(In)xBy’’ materials with a stoichio-metry of 1 : 6 using the CALYPSO code. Experimentally, not onlythe global minimum energy structure can be fabricated, butalso some metastable structures have been successfullysynthesized.52,53,78 Through fundamental structural searching,we obtain one type of rectangular structure [Pnmm (no. 47)]with the global minimum energy and another rectangularstructure [Pmm2 (no. 25)] (see Fig. S1, ESI†) with the secondminimum energy. The energy of this metastable structure ishigher by 71.4 and 12.9 meV per atom than that of the globalminimum rectangular GaB6 and InB6, respectively. Besides, ahexagonal structure [P6/mmm (no. 191)] with the third lowesttotal energy for 2D Ga(In)B6 is also obtained. To study thedifferent phases in these systems, in the following works, weonly focus on the first rectangular structure and the hexagonalstructure. For simplicity, we name them as rect- and hex-XB6

(X = Ga, In), respectively (see Fig. 1). We further calculate theircohesive energy (Ecoh) and formation energy (Ef) by

Ecoh ¼ EX þ 6EB � EXB6

7(7)

and

Ef ¼EXB6

� mX � 6mB7

(8)

where EXB6is the total energy of one formula unit of XB6

(X = Ga,In), and EX and EB are the energies of isolated X andB atoms, respectively. mX is the chemical potential of bulkGa(In) and mB is the chemical potential of w3 borophene. Ecohand Ef are useful parameters to evaluate whether a predicted 2Dmaterial is possible to synthesize experimentally.57,79 The Ecohvalues of rect-, hex-GaB6, rect-, and hex-InB6 are estimated to be5.01, 4.82, 4.78 and 4.57 eV per atom, respectively. The Ef valuesof rect-, hex-GaB6, rect-, and hex-InB6 are estimated to be�0.77,�0.96, �0.97 and �1.20 eV per atom, respectively, which arelarger than those of tri-FeB6 (�0.46 eV per atom), a-FeB6

(0.07 eV per atom) b-FeB6 (�0.04 eV per atom) and rect-AlB6

(�0.45 eV per atom), indicating their exothermic properties andthat they are likely to be synthesized under certain experi-mental conditions. As shown in Fig. 1, both rect- and hex-XB6

structures are stacked with three sublayers in the order ofB–Ga(In)–B. Two distorted boron honeycombs are connectedby the metal sublayer in rect-XB6 (see Fig. 1a and b), while tworeal planar boron honeycombs are connected by the metalsublayer in hex-XB6 (see Fig. 1c and d). Meanwhile, the metallayer is stabilized by the two borophene sheets on eitherside, providing high stability to the complete 2D structures.In rect-XB6, there are three symmetrically distinct atom types:one for Ga(In) and two for B (labled as B1 and B2). In hex-XB6,there are one type of Ga(In) and one type of B atoms. Afteroptimization, their lattice parameters and bond lengths arelisted in Table 1, which are comparable to the reported AlB6

systems.17 The B–B bonds are all shorter than the Ga(In)–B andGa–Ga(In–In) bonds, indicating that the stability and themechanical properties of these structures are mainly governedby the B–B bonds. The Ga–Ga(In–In) bonds are too long to formchemical bonding.

3.2 Stability

The stability of 2D crystals is important in predicted structures.Here, we study their thermal stability by performing AIMDsimulations. We adopt a 3 � 3 � 1 supercell to minimize theeffects of periodic boundary conditions. The results of the variationof free energy in the AIMD simulations within 10 ps, along with thelast frame of the photographs, are exhibited in Fig. 2. rect-GaB6 isthe most stable among our systems, maintaining structuralintegrity up to 1000 K, which is also higher than that of hex-W2B2 (900 K).

60 Such high thermal stability of rect-GaB6 will make

Table 1 Lattice parameters and bond lengths for rect-, hex-GaB6, rect-, and hex-InB6. There are two types of B1–B1 in rectangular structures, one ofthem bonds with the side shared by hexagonal holes (labled as B1–B1) and the other bonds along the diagonal of the rhombus (labled as B1–B10). Themetal bonding along the b direction is indicated by the data within parentheses

Compounds a (Å) b (Å) B1–B1 (Å) B1–B10 (Å) B1–B2 (Å) Ga(In)–B1 (Å) Ga(In)–B2 (Å) Ga–Ga(In–In) (Å)

rect-GaB6 2.93 3.41 1.693 1.723 1.716 2.501 2.333 2.925(3.415)hex-GaB6 3.42 3.42 1.711 2.426 3.422rect-InB6 2.91 3.42 1.717 1.749 1.725 2.698 2.580 2.951(3.466)hex-InB6 3.46 3.46 1.730 2.628 3.459

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it easy to be fabricated and explored in experiments. hex-GaB6

and rect-InB6 are both stable at 500 K, and hex-InB6 only retainsintegrity under 300 K. Overall, our structures are all stable atroom temperature and they satisfy the thermal stability.

To further evaluate the stability of these structures, wecalculate the elastic constants Cij by

Es ¼1

2C11exx2 þ

1

2C22eyy2 þ C12exxeyy þ 2C66exy2; (9)

where Es is strain energy and the tensile strain is defined as

e ¼ a� a0

a0. Here, a and a0 are the lattice constants of the

strained and strain-free structures, respectively. In order tocalculate the elastic stiffness constants, Es as a function of eat strain of �2.5% r e r 2.5% with an increment of 0.25% iscalculated. Then, Cij can be obtained using the VASPKIT,80 apost-processing program for the VASP code. Based on theresults of Cij, the in-plane Young’s modulus (Y) along the x

and y direction can be calculated by Yx ¼ C11C22 � C122

C22and

Yy ¼C11C22 � C12

2

C11, respectively. The Poisson’s ratio can be

calculated by nx ¼ C12

C22and ny ¼

C12

C11. Thses results are listed

in Table 2. The stability criteria for rect-XB6 are C11C12–C122 4 0

and C66 4 0,17,81 while the stability criteria for hex-XB6 areC11 4 |C12| and C66 4 0.81,82 All our crystals satisfy thesecriteria, indicating they are mechanically stable. The values oftheir Young’s modulus are much larger than those of M2C3

(88.56, 65.68, and 51.81 N m�1) (M = As, Sb, Bi),83 silicene(62.31 N m�1), MoS2 (123 N m�1),84 tetr-W2B2 (152 N m�1), andhex-W2B2 (271 N m�1),60 and are comparable to graphene(340 N m�1).85 Besides, the values of Young’s modulus of rect-GaB6 are equal to those of 2D rect-AlB6 (437 and 379 N m�1),17

while those of the other systems are all smaller than them. ThePoisson’s ratios for rect-, hex-GaB6 and hex-InB6 are smallerthan those of tetr-W2B2 (0.69)

60 and 2D WB2 (0.67),86 while the

hex-InB6 system is comparable to that of hex-W2B2 (0.22).60

Moreover, the Poisson’s ratios in our predicted structures arelarger than those of the 2D rect-AlB6 (0.06).17 Meanwhile,hex-GaB6 and hex-InB6 are isotropic with identical Poisson’sratios along x and y directions, while rect-GaB6 and rect-InB6 areanisotropic.

3.3 Electron properties

To gain insights into the bonding nature, we calculate thedifference charge density, as shown in Fig. 3. We can see clearlythat the charges transfer from metal atoms to the B atoms.Meanwhile, charges transfer a little among B atoms. This isunderstandable because B atoms are electrically neutral. Wealso calculate the line charge density distribution and performthe Bader analysis.87 The results of the Bader charge (QB) andcharge density at the corresponding bond point (CDb) analysesare listed in Table 3. Here, we can further conclude that the B1

Fig. 2 Variation of the free energy in the AIMD simulations during thetime scale of 10 ps along with the last frame of photographs for (a) rect-,(b) hex-GaB6, (c) rect- and (d) hex-InB6, respectively.

Table 2 Calculated parameters of Cij (N m�1), Young’s modulus (N m�1)and the Poisson’s ratio for rect-, hex-GaB6, rect- and hex-InB6

Compounds C11 C12 C22 C66 Yx Yy nx ny

rect-GaB6 440 34 380 145 437 378 0.09 0.08hex-GaB6 343 105 343 119 311 311 0.31 0.31rect-InB6 234 46 355 18 228 346 0.13 0.09hex-InB6 394 68 394 163 382 382 0.17 0.17

Fig. 3 Top and side views of difference charge density for (a and b) rect-,(c and d) hex-GaB6, (e and f) rect-, and (g and h) hex-InB6. The yellow andcyan areas represent electron gains and losses, respectively.

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atoms provide partial charges to B2 atoms in coordination withGa(In) atoms in rect-XB6, while the B atoms gain electroncharges only from Ga(In) atoms in hex-XB6. Comparing thevalues of CDb with a typical covalent compound Si (0.104 ea.u.�3) and a typical ionic compound NaCl (0.007 e a.u.�3),88

here, the B–B bonds are found to exhibit strong covalentproperties within the B atomic layer while the Ga(In)–B bondspossess mixed features of ionic and covalent bonds. By analyzingthe ionicity according to the Bader charges, the ionic charges ofthese crystals are found to be different. For rect-GaB6, the chargestransferring from B1 to B2, B10 to B2, and Ga to B2 are 0.08, 0.2and 0.804 electrons per atom, respectively. As for hex-GaB6, it canbe represented as hex-Ga�0.793B6

+0.132. More details for the InB6

systems can be found in Table 3. These results accord well withthe distribution of the difference charge density (see Fig. 3). It isclear that the electron transfer of hex-GaB6 is the most dominantone among all our predicted 2D crystals, even more dominantthan that of the 2D superconductor Mo2B2.

51

In order to study the electron properties of our predictedstructures, we calculate the orbital-resolved band structures as

well their corresponding electronic DOS, as exhibited in Fig. 4.The high-symmetry paths of rect- and hex-XB6 are along G–Y–S–X–G–S and G–M–K–G, respectively. At first sight, the distributionsof the band structures for rect-XB6 are also similar (see Fig. 4aand c) to hex-XB6 (see Fig. 4c and d). However, we can find somedifferences that the orbits of In atomsmove downward comparedwith Ga, due to their different atomic mass. By calculating theband structures including the spin–orbit couping (SOC) (notshown), we demonstrate that the SOC has little effect on oursystems. Thus, we will not include it in our following calcula-tions. From our calculated band structures, it is easy to find thatall our obtained crystals are intrinsic metals with some bandscrossing the Fermi level. We further calculate their correspondingFermi surfaces and present them in Fig. 5. We can see that theyare all in accordance with the band distributions at the Fermienergy level. For rect-GaB6, as presented in Fig. 4a and 5a, thereare eleven bands crossing the Fermi level: two along G–Y, threealong Y–S, two along S–X, one along X–G and three along G–S.As indicated in Fig. 4c and 5c, due to the downward orbits ofrect-InB6, there are more bands that cross the Fermi level as

Table 3 Bader charge (QB) and charge density at the corresponding bond point (CDb) for (a and b) rect-, (c and d) hex-GaB6, (e and f) rect-, and (g and h)hex-InB6

Compounds QB (B1) QB (B10) QB (B2) QB (Ga(In)) CDb (B1–B1)/(B1–B10)/(B1–B2) CDb (Ga(In)–B1)/(Ga(In)–B2)

rect-GaB6 2.92 2.80 3.681 2.196 0.143/0.121/0.13 0.040/0.051rect-InB6 2.91 2.94 3.560 2.172 0.137/0.115/0.129 0.031/0.036hex-GaB6 3.132 2.207 0.128 0.046hex-InB6 3.145 2.133 0.124 0.034

Fig. 4 Band structures as well as DOS for (a and b) rect-, (c and d) hex-GaB6, (e and f) rect-, and (g and h) hex-InB6. The Fermi level is set as zero.

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compared with rect-GaB6. As for hex-GaB6, three bands crossthe Fermi level (see Fig. 4b), as illustrated by the Fermi surface(see Fig. 5b). The first band contributes the hexagram of theFermi sheet surrounding the G center and the other two bandsform two types of arc around the vertexes. While compared withhex-GaB6, the Fermi surface of hex-InB6 changes a lot becausethe second band crossing the Fermi level is replaced. From ourcalculated band structures as well as the electronic DOSs, it isclear that the B-2p orbits make the dominant contributionaround the Fermi level, followed by a partial contribution fromGa(In)-s orbits. Thus, we can conclude that the metallic natureof our obtained crystals is controlled mainly by their B-2porbits, which is the same as 2D AlB6 systems.17

3.4 EPC and surpercondutivity

The recently reported 2D superconductors17,51,60,75,77,89–91 stimulatedus to study the possible superconductivity in our present metallicsystems. The results of phonon spectra, the magnitude of theEPC lqn, the phonon density of states (PhDOS), the Eliashbergelectron–phonon spectral function a2F(o) and the cumulativefrequency dependent EPC l(o) are shown in Fig. 6. As indicatedin Fig. 6a, e, i and m, it is clear that our predicted structures areall dynamically stable with the absence of imaginary frequencies.From the decomposed phonon spectra along with PhDOS, we canfind that the Ca(In) atomic vibrations mostly appear in the low-frequency region due to their heavy atomic mass. They dominatethe three acoustic branches while the B atomic vibrations occupythe optical modes. Besides, the highest phonon frequencies forthem are all above 1100 cm�1, which is comparable to borophene(1274 cm�1),75 indicating strong bonding interactions between Band B atoms. From our calculated l(o) and a2F(o) for rect-GaB6,as shown in Fig. 6d, we find that the phonons in the regionof 200–400 cm�1, which are mainly from the out-plane modes

of B atoms, make more than half the contribution of the totalEPC (l = 0.39), about 55%. Obviously, the large peaks of a2F(o)appear and lead to an increase for l(o) in this range. Theseresults also accord with the distributions of lqn (see Fig. 6b).Other phonon modes contribute to the remaining EPC. Based onthe McMillan–Allen–Dynes formula,72 its superconducting transi-tion temperature Tc is calculated to be 1.67 K. As for hex-GaB6

(see Fig. 6h), the out-plane modes of B atoms between 200 and400 cm�1 account for 61.54% of the total EPC (l = 0.65), whichaccords with the large values of lqn around the M point (seeFig. 6f). In this range, the two large peaks of a2F(o) are obvious,resulting in the fact that l(o) increases rapidly in this range andthe Tc increases to 14.02 K. Besides, as shown in Fig. 6l and p,rect- and hex-GaB6 are also found to be intrinsic superconductorswith a Tc of 7.77 and 4.83 K, respectively. However, they are allweak superconductors due to their l o 1.

In Table 4, we compare the superconductive parameters ofour predicted superconductors with some recently reported 2Dintrinsic boride superconductors. One can see that the Tc ofhex-GaB6 is the highest among our predicted structures, and itis even higher than most of the boride superconductors,such as Li2B7,

91 2D Mo2B251 and 2D W2B2.

60 In fact, the Tc ofhex-GaB6 is higher than that of B allotrope (a sheet)89 and 2Drect-AlB6,

17 but it is still lower than those of borophene (b12 andw3)

73,75 and B2C.90 This is understandable because the vibrations

of B atoms are constrained in our systems like most borides. It isnoteworthy that increasing the value of m will lead to a decrease ofTc. We do not want to discuss this effect in detail.

As we know, strain is usually used to modulate the electronicproperties of 2D systems and can enhance/suppress the super-conducting properties.17,60,75,92–94 In our present study, wewant to discuss to some extent the effects of biaxial strain onthe superconduting properties of our predicted 2D materials.

The biaxial strain (x) is calculated by x ¼ a� a0

a0� 100%

(a positive value means tensile strain while a negative onemeans compressive strain). The atomic coordinates are fullyrelaxed for each case. The changes in their superconductiveparameters [N(EF), olog, l and Tc] under a series of strains areexhibited in Fig. 7. Here, the range of strain for each crystal isthe largest dynamically stable region. For rect-GaB6, the N(EF)decreases first to 6.33 states per spin per Ry per cell and then itrebounds to increase at a strain of �3%. The value of olog

monotonically increases when strain changes from �4% to 2%,while l and Tc decrease in the full range of strain. Obviously,compressive strain is beneficial to the superconductivity of rect-GaB6. Its Tc can be increased to 6.9 K at a strain of �4%. As forhex-GaB6 (see Fig. 7b), N(EF) and olog change in the oppositedirection. N(EF) increases first and then decreases at a strain of�3%, reaching its maximum value of about 8.07 states per spinper Ry per cell. In contrast, olog has its minimum value of343.88 K at a strain of �3%. Meanwhile, the trends of l and Tcare similar to N(EF). Under compressive strain of 3%, Tc reachesits maximum value of 22.23 K with l = 0.96. As shown in Fig. 7c,we can see that rect-InB6 is only dynamically stable withina tensile strain of 2% and the strain can suppress its

Fig. 5 Fermi surfaces for (a) rect-, (b) hex-GaB6, (c) rect-, and (d) hex-InB6. The Fermi energy level is set as zero.

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superconductivity. Its highest Tc is 7.77 K without appliedstrain. The superconductivity of hex-InB6 is sensitive to strain,as indicated in Fig. 7d. Under a compressive strain of 2%, its Tcis enhanced to 19.23 K with l = 0.82 and then decreases untilthe tensile strain is applied. In general, compared with thefreestanding state, the compressive strain and tensile strain canlargely enhance the Tc of hex-InB6, while the tensile strain cansuppress the Tc of the other systems.

4 Substrates

Recently, some 2D materials have been successfully preparedon suitable substrates.4,95–100 Therefore, we also want our pre-dicted systems of rect- and hex-XB6 (X = Ga, In) to be synthesizedon substrates. In our present work, we theoretically select Cu andPt as ideal substrates. The substrate-supported structures of rect-,hex-GaB6, rect-, and hex-InB6 are shown in Fig. 8. To simulate the

Fig. 6 Phonon dispersion, lqn, PhDOS, a2F(o) and l(o) of (a–d) rect-, (e–h) hex-GaB6, (i–l) rect-, and (m–p) hex-InB6. The red, blue, pink, orange, darkred and yellow hollow circles in (a, e, i and m) represent B horizontal, B vertical, Ga horizontal, Ga vertical, In horizontal and In vertical modes,respectively. The magnitude of lqn is displayed with an identical scale in all figures for comparison.

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effects of substrates, we fix their atomic coordinates for eachrelaxation. The adsorption energy is calculated by

Eads ¼Emolþslab � ðEmol þ EslabÞ

n; (10)

where Emol+slab denotes the total ground state energy of theoptimized configuration of our system adsorbed on the slabsurface, Eslab is the total energy of the slab with a clean surface,Emol is the energy of our system and n is the atom number. Thelattice mismatch (d) is calculated by

d ¼ Lmol � Lslab

Lslab; (11)

where Lmol and Lslab are the corresponding lattice parameters ofthe 2D crystal and the slab, respectively. The results of ourcalculated adsorption energy and lattice mismatch are presentedin Table 5. The negative values of the adsorption energiesindicate that our predicted structures are suitable to be preparedon such substrates. Meanwhile, the big adsorption energy andthe small lattice mismatch are beneficial to practically grow ourcrystals in experiments. Thus, according to our calculated resultsof Eads and d, we recommend that rect-GaB6 and rect-InB6 shouldbe grown on the Cu(100) surface while hex-GaB6 and hex-InB6

should be grown on the Pt(111) surface.

5 Conclusion

In conclusion, we have predicted two types of 2D XB6 (rect- andhex-XB6) (X = Ga, In), by combining the crystal structureprediction technique and the first-principles method. Aftertesting their thermal, dynamical, and mechanical stabilities,we systematically studied their bonding types, electron andphonon properties, and superconductivity. According to theBardeen–Cooper–Schrieffer theory, we have demonstrated thatthe rect-, hex-GaB6, rect- and hex-InB6 systems are all intrinsicphonon-mediated superconductors with Tc values of 1.67,14.02, 7.77 and 4.83 K, respectively. The in-plane vibrations ofthe B atoms and the B-2p electronic orbitals play crucial roles intheir superconductivity. The inter-layered metal atoms contributeelectrons to the covalent B layers, making the Tc of hex-GaB6higher than those of B allotrope (a sheet),89 Li2B7,

91 2D Mo2B2,51

2D W2B2,60 and 2D rect-AlB6.

17 We simulated their super-conductivity under applied biaxial strains and recommendedthat rect-XB6 should be grown on the Cu(100) surface whilehex-XB6 should be grown on the Pt(111) surface. Our interesting

Table 4 Superconductive parameters of m*, N(EF) (in the unit of states perspin per Ry per cell), o (in K), l and Tc (in K) for some intrinsic 2D boridesuperconductors

Compounds m* N(EF) olog l Tc Ref.

B2C 0.1 314.8 0.92 19.2 90Li2B7 0.1 462.8 0.56 6.2 91B allotrope 0.05 5.85 262.2 0.52 6.7 89B (b12) 0.1 8.12 425 0.69 14 75B (w2) 0.1 323.43 0.95 24.7 73tetr-Mo2B2 0.1 16.02 344.84 0.49 3.9 51tri-Mo2B2 0.1 16.81 295.0 0.3 0.2 51tetr-W2B2 0.1 12.46 232.4 0.69 7.8 60hex-W2B2 0.1 13.60 232.2 0.43 1.5 60rect-AlB6 0.05 6.41 345.2 0.36 4.7 17rect-GaB6 0.1 6.40 490.3 0.39 1.67 This workhex-GaB6 0.1 7.79 436.1 0.68 14.02 This workrect-InB6 0.1 8.69 471.2 0.55 7.77 This workhex-InB6 0.1 9.43 154.1 0.67 4.83 This work

Fig. 7 Trends of superconductive parameters for (a) rect-, (b) hex-GaB6,(c) rect-, and (d) hex-InB6 under a series of strains.

Fig. 8 Side views of rect-GaB6 on (a) Cu(100) and (e) Pt(100), rect-InB6 on(b) Cu(100) and (f) Pt(100), hex-GaB6 on (c) Cu(111) and (g) Pt(111), and hex-InB6 on (d) Cu(111) and (h) Pt(111). The red orange, yellow, blue andchartreuse balls represent B, Ga, In, Cu and Pt atoms, respectively.

Table 5 Adsorption energy Eads and the lattice mismatche d for rect- andhex-GaB6, and rect- and hex-InB6 on Cu and Pt surfaces

rect-GaB6 Eads (eV) d (%) hex-GaB6 Eads (eV) d (%)

Cu(100) �0.48 2.2 Cu(111) �3.54 5.4Pt(100) �0.26 8.3 Pt(111) �4.29 1.5

rect-InB6 Eads (eV) d (%) hex-InB6 Eads (eV) d (%)

Cu(100) �0.58 0.9 Cu(111) �3.51 6.5Pt(100) �0.09 7.0 Pt(111) �3.78 0.4

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results in the present work provide a choice and also a mechanismin realizing superconduction in metal borides and may stimulatefurther efforts in the field of 2D superconducting materials.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors would like to acknowledge the financial supportfrom the National Natural Science Foundation of China underGrant No. 11835008, and 51872250, the State Key Laboratory ofIntense Pulsed Radiation Simulation and Effect (NorthwestInstitute of Nuclear Technology) under Grant No. SKLIPR1814,and the Key Laboratory of Low Dimensional Materials & ApplicationTechnology of Ministry of Education (Xiangtan University) underGrant No. KF20180203. P. F. L. and B. T. W. also acknowledge theGuangdong Provincial Department of Science and Technology,China (No. 2018A0303100013) and the Program of State KeyLaboratory of Quantum Optics and Quantum Optics Devices (No.KF201904). The calculations were performed at SupercomputerCentre at the China Spallation Neutron Source.

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