Luo
Yan
^{abcd},
Tao
Bo
^{cd},
Wenxue
Zhang
^{ef},
Peng-Fei
Liu
^{cdef},
Zhansheng
Lu
^{g},
Yong-Guang
Xiao
*^{ab},
Ming-Hua
Tang
*^{ab} and
Bao-Tian
Wang
*^{cdef}
^{a}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.cn; mhtang@xtu.edu.cn
^{b}Hunan Provincial Key Laboratory of Thin Film Materials and Devices, School of Material Sciences and Engineering, Xiangtan University, China
^{c}Institute of High Energy Physics, Chinese Academy of Sciences (CAS), Beijing 100049, China. E-mail: wangbt@ihep.ac.cn
^{d}Dongguan Institute of Neutron Science (DINS), Dongguan 523808, China
^{e}Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, 030006, China
^{f}State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan, 030006, China
^{g}College of Physics and Materials Science, Henan Normal University, Xinxiang 453007, China
First published on 19th June 2019
Two-dimensional (2D) superconductors, which can be widely applied in optoelectronic and microelectronic devices, have gained renewed attention in recent years. Based on the crystal structure prediction method and first-principles calculations, we obtain four novel 2D tungsten boride structures of tetr-, hex-, and tri-W_{2}B_{2} and hex-WB_{4} and investigate their bonding types, electronic properties, phonon dispersions and electron–phonon coupling (EPC). The results show that both tetr- and hex-W_{2}B_{2} are intrinsic phonon-mediated superconductors with a superconducting transition temperature (T_{c}) of 7.8 and 1.5 K, respectively, while tri-W_{2}B_{2} and hex-WB_{4} are normal metals. We demonstrate that carrier doping as well as biaxial strain can soften the low-frequency phonon modes and enhance the strength of the EPC. While the T_{c} of tetr-W_{2}B_{2} can be increased to 15.4 K under a compressive strain of −2%, the T_{c} of hex-W_{2}B_{2} can be enhanced to 5.9 K by a tensile strain of +4%. With the inclusion of spin–orbit couping (SOC), the value of T_{c} decreases by 38.5% in our systems. Furthermore, we explore the stabilities and mechanical properties of tetr- and hex-W_{2}B_{2} and indicate that they may be prepared by growing on ZnS(100) and ZnS(111), respectively. Our findings provide novel 2D superconducting materials and will stimulate more efforts in this filed.
In recent years, rapid advancements in nanotechnology, including molecular beam epitaxy,^{13} atomic layer deposition,^{14} pulsed laser deposition,^{15}in situ ultrahigh vacuum-low temperature measurements,^{16} scanning tunneling microscopy/spectroscopy^{17}etc., have greatly improved the studies on ultrathin 2D superconductors. For example, the interface between LaAlO_{3} and SrTiO_{3}, both of them being perovskite oxide insulators, exhibits 2D superconductivity.^{18} An electric field has been found to induce 2D superconductivity in insulating materials of SrTiO_{3} and ZrNCl.^{19,20} For a 1 unit cell thick FeSe layer, epitaxially grown on SrTiO_{3} substrates, its T_{c} was found to reach up to 40–100 K.^{21,22} Another typical system is graphene,^{23,24} which can be modulated to exhibit superconductivity via Ca-intercalation^{25,26} and Li-decoration.^{27,28} Moreover, the superconductivity of borophenes and monolayers of transition-metal dichalcogenides is also under rapid exploration.^{29–33} In brief, the study of 2D superconductivity has evolved into one of the most active fields, closely relating to material science, device physics, instrumental technologies and fundamental physics.^{1}
Till now, detailed studies of the superconducting properties of 2D transition metal borides (TMBs) have been scarce from both experimental and theoretical aspects,^{1,34,35} although they have been widely studied due to their unique mechanical and electronic properties.^{36–41} The harsh synthesis conditions^{34} may have greatly limited their developments and characterizations. Like other 2D superconductors,^{42–45} 2D TMBs may also be applied in optoelectronic and microelectronic devices. Thus, it is urgently needed to design new 2D TMBs and explore their superconductivity. Generally, to reveal the underlying mechanism of a newly discovered superconductor, besides the experimental measurements, another effective way is to perform electron–phonon coupling (EPC) calculations according to the Bardeen–Cooper–Schrieffer (BCS)^{46} theory, which has been proved to be successful in many systems.^{25–28,31,32,47,48}
In the present work, we obtain four novel stable structures of 2D tungsten borides (TBs) (labled as tetr-, hex-, and tri-W_{2}B_{2} and hex-WB_{4} and presented in Fig. 1) and perform first-principles calculations to study their bonding types, electronic properties and superconductivity. We find that tetr- and hex-W_{2}B_{2} are intrinsic superconductors with T_{c} of 7.8 and 1.5 K, respectively, while tri-W_{2}B_{2} and hex-WB_{4} don't exhibit superconducting properties. The value of T_{c} decreases by 38.5% when taking the spin–orbit couping (SOC) effects into consideration on our obtained superconductors. The superconducting properties of tetr- and hex-W_{2}B_{2} can be effectively tuned by carrier doping and biaxial strain. The mechanical, dynamical, and thermal stabilities of these two new superconducting phases are illustrated by our calculated elastic constants, phonon spectra, and our ab initio molecular dynamics (AIMD) simulations. After comparing substrates of SiC, CuI, and ZnS, we find that the ZnS(100) and ZnS(111) substrates may be suitable for tetr- and hex-W_{2}B_{2}, respectively.
The electronic structures and superconducting properties are calculated by using plane-wave code QUANTUM ESPRESSO (QE).^{58} The norm-conserving GGA pseudopotentials^{59} are used to model the electron–ion interactions. After the full convergence tests, the plane-wave kinetic-energy cutoff and the charge-density cutoff are chosen as 80 and 320 Ry, respectively. Brillouin zone (BZ) integrations of tetr-, hex-, and tri-W_{2}B_{2} and hex-WB_{4} are all sampled on a 32 × 32 × 1 k-point grid, with a Methfessel–Paxton smearing width of 0.02 Ry. The lattice constants as well as the atomic sites are fully optimized. The dynamic matrix and EPC matrix elements are calculated on a q-mesh of 16 × 16 × 1 for tetr- and hex-W_{2}B_{2} and 4 × 4 × 1 for tri-W_{2}B_{2} and hex-WB_{4}. The phonon modes and EPC are calculated using the density-functional perturbation theory (DFPT)^{60} and the Eliashberg equation.^{61,62} The T_{c} is evaluated by using the Allen–Dynes modified McMillan formula^{63} with a typical Coulomb pseudopotential of μ* = 0.1.^{64–66}
To gain more insights on these new structures of W–B systems, we calculate the charge density as well as the difference charge density and plot them in Fig. 2. Here, the difference charge density is calculated by subtracting the densities of noninteracting component systems, ρ(W) + ρ(B), from the density of TBs. We also calculate the line charge density distribution along the nearest B–B, W–W, and W–B bonds and perform the Bader analysis.^{72} The results of the Bader charge (Q_{B}), bond lengths, and charge density at the corresponding bond point (CD_{b}) are listed in Table 1. As a general rule, when the value of CD_{b} is smaller than 0.007 e a.u.^{−3} found for the Na–Cl bond in the typical ionic crystal of NaCl,^{73} we believe the bonding is ionic. When the value of CD_{b} is higher than 0.104 e a.u.^{−3} found for the Si covalent bond,^{73} we believe the bonding is covalent. If it is in between 0.007 and 0.104 e a.u.^{−3}, we would state that the bonding possesses mixed features of ionic and covalent bonding.
Compounds | Q _{B} (W) | Q _{B} (B) | B–B | W–W | W–B | CD_{b} (B–B) | CD_{b} (W–W) | CD_{b} (W–B) |
---|---|---|---|---|---|---|---|---|
tetr-W_{2}B_{2} | 5.16 | 3.84 | 3.07 | 3.07 | 2.19(2.62) | 0.046 | 0.045 | 0.081(0.070) |
hex-W_{2}B_{2} | 5.49 | 3.51 | 1.76 | 3.05 | 2.28 | 0.113 | 0.043 | 0.073 |
tri-W_{2}B_{2} | 5.67 | 3.33 | 1.75 | 2.85 | 2.18 | 0.118 | 0.050 | 0.064 |
hex-WB_{4} | 5.69 | 3.31 | 1.71 | 2.97 | 2.37 | 0.128 | 0.044 | 0.061 |
Based on Table 1 and Fig. 2, we can deduce the following characters for tetr-W_{2}B_{2}: (i) the two typical W–B bond lengths of about 2.4 Å are greatly shorter than those of the W–W (3.07 Å) and B–B (3.07 Å) bonds, indicating that the stability and the mechanical properties of this structure are mainly governed by the W–B bonds; (ii) the W–B bonds show mixed features of ionic and covalent characters with CD_{b} (W–B) = 0.070 (0.081) e a.u.^{−3}.
For hex-W_{2}B_{2}, B atoms connect with each other by strong covalent bonds within the B atomic layer with CD_{b} (B–B) = 0.113 e a.u.^{−3} while the bonding between B and W layers is relatively weak. As for tri-W_{2}B_{2}, its bonding features are quite similar to that of hex-W_{2}B_{2}, but with shorter W–B and W–W bonds. This answers to some extent why tri-W_{2}B_{2} is more stable. Considering the strong covalent bonds with CD_{b} (B–B) = 0.118 e a.u.^{−3} in tri-W_{2}B_{2}, one may wonder why it is not the energetically most stable one. In fact, there are ten W–B bonds in one unit cell of tetr-W_{2}B_{2} while there are only one B–B bond and four W–B bonds in tri-W_{2}B_{2}. Thus, the fact that tetr-W_{2}B_{2} is more stable than tri-W_{2}B_{2} is understandable. For hex-WB_{4}, B atoms bond with each other with very short covalent bonds; W atoms bond with each other with weak metal bonds; the adjacent W and B layers are bonded by W–B bonds with mixed features of ionic and covalent bonding.
From our calculated difference charge density, we can see that the electrons are accumulated at each B atom from the vertical direction of the W atoms, indicating electron transfer from W to B. By analyzing the ionicity according to the Bader charges, the ionic charges of these TBs are different. For simplicity, their ionic charges can be represented as tetr-W_{2}^{0.84−}B_{2}^{0.84+}, hex-W_{2}^{0.51−}B_{2}^{0.51+}, tri-W_{2}^{0.33−}B_{2}^{0.33+} and hex-W^{0.31−}B_{4}^{0.31+}, respectively. It is easy to conclude that the electron transfer of tetr-W_{2}B_{2} is the most dominant among all our predicted 2D TBs, even more dominant than that of tetr-Mo_{2}B_{2}, which has been reported to be a superconductor.^{74}
Besides, we calculate the electron localization functions (ELFs)^{75} and plot their three-dimensional (3D) and 2D perspectives in Fig. 3. In general, regions with ELF values close to 1 have strong covalent bonding electrons or lone-pair electrons, regions with ELF values close to 0 are typical of very low electron density, and regions with ELF values close to 0.5 are typical for a homogeneous electron gas. As shown in Fig. 3(a), large values of ELF between W–B bonds and small values between W–W and B–B bonds are clear. This indicates that the W–B bonds in tetr-W_{2}B_{2} are the strongest, consistent with our previous charge density analyses. As for hex-W_{2}B_{2}, tri-W_{2}B_{2}, and hex-WB_{4}, B–B and W–B bonds are found with large values of ELF while there are almost no interactions in W–W [see Fig. 3(b)–(d)].
Fig. 3 ELF planes of (a) tetr-, (b) hex-, and (c) tri-W_{2}B_{2} and (d) hex-WB_{4} viewed in 3D (top panels) and 2D (bottom panels). The ELF ranges from 0 to 1 are indicated by the color bar in (c). |
Fig. 4 shows the orbital-resolved band structures and electronic density of states (DOS) without SOC and with SOC. The corresponding Fermi surfaces without SOC and with SOC are presented in Fig. 5. We find that all our predicted TBs exhibit intrinsic metallic features with many bands crossing the Fermi level. For tetr-W_{2}B_{2}, there are three bands crossing the Fermi level [see Fig. 4(a)]. As shown in Fig. 5(a), the first band contributes the elliptical hole Fermi sheet surrounding the Γ center. Dissimilarly, the second band forms a four-leaf clover type Fermi sheet. The third band forms a butterfly type Fermi sheet around the BZ boundary. For hex-W_{2}B_{2}, as presented in Fig. 4(d) and 5(c), there are three and four bands crossing the Fermi level along Γ–M and Γ–K, respectively. The first band forms a hexagonal Fermi sheet around the Γ center and it is surrounded by a round Fermi sheet formed by the second band. Besides, the ellipical and the trapezoidal Fermi sheets formed by other bands alternately distribute around the BZ boundary. For tri-W_{2}B_{2}, there are three bands crossing the Fermi level [see Fig. 4(h)]. The first band forms a hexagonal Fermi sheet around the Γ center while the second and the third bands contribute the semielliptical Fermi sheets around the BZ boundary. For hex-WB_{4}, there are one and three crossings along the M–K and Γ–K paths, respectively [see Fig. 4(j)]. These crossings form two Dirac cones, one at K and another half way along Γ–K, near the Fermi energy level. The inclusion of SOC leads to the degeneration of bands in the whole BZ for 2D W_{2}B_{2}. For hex-W_{2}B_{2} and hex-WB_{4} monolayers, SOC introduces gaps in the massless Dirac fermion, as shown in Fig. 4(b) and (d). Meanwhile, a continuous gap extends over the whole BZ of tri-W_{2}B_{2} when SOC is taken into account. These results are in accordance with the Fermi surfaces shown in Fig. 5. From our calculated electronic DOS for the three phases of W_{2}B_{2}, we can see that the W-5d orbitals contribute dominantly around the Fermi energy level while the contribution from the B-2p orbitals is limited. For hex-WB_{4}, the B-2p orbitals dominate near the Fermi level with some contributions from the W-5d orbitals. Thus, we can conclude that the metallic nature of these 2D W_{2}B_{2} structures is mainly controlled by their W-5d orbitals while that of hex-WB_{4} is governed by the B-2p orbitals.
Fig. 5 Fermi surfaces without SOC (left one) and with SOC (right one) of (a and b) tetr-, (c and d) hex-, (e and f) and tri-W_{2}B_{2} and (h and i) hex-WB_{4}. |
In order to explore the possible superconductivity of our obtained 2D TBs, we calculate their phonon spectra, phonon density of states (PhDOS) and electron–phonon coupling (EPC). The results of their phonon dispersions along the high-symmetry paths and PhDOS are displayed in Fig. 6. It is clear that all our predicted TBs are dynamically stable with the absence of imaginary frequencies in their BZs. From the decomposition of those phonon spectra with respect to the W and B atomic vibrations as well as the partial PhDOS, we find that the W atomic vibrations dominate the three acoustic branches while the B vibrations mainly occupy the optical modes. Besides, for hex-W_{2}B_{2}, the highest phonon frequency is 920 cm^{−1}, which is the highest in our obtained TBs and also higher than that of Mo_{2}B_{2} (880 cm^{−1})^{74} but smaller than that of borophene (1274 cm^{−1}).^{76} Such a high frequency indicates strong bonding interactions between the B and B atoms in hex-W_{2}B_{2}.
The phonon dispersions can be weighted by the magnitude of the EPC λ_{qν}. According to the Migdal–Eliashberg theory,^{77} the λ_{qν} is calculated by
(1) |
(2) |
(3) |
(4) |
(5) |
From our calculated cumulative frequency dependence of EPC λ(ω) for tetr-W_{2}B_{2}, we find that the phonons below ∼200 cm^{−1} account for 0.54 (78%) of the total EPC (λ = 0.69), phonons in the intermediate region of 330–500 cm^{−1} account for 17% while in the high-frequency region, they contribute the remaining 5%. Obviously, the large values of λ_{qν} along the Γ–M–X directions in the frequency range of 65–130 cm^{−1} result in two large peaks on the PhDOS and α^{2}F(ω). As a consequence, λ(ω) increases rapidly in this frequency range and the large values of λ_{qν} here are the main origin of its superconductivity.
As for hex-W_{2}B_{2}, the W atomic vibrations (W_{z} and W_{xy} modes) dominate the low-frequency region (below 200 cm^{−1}) while the interactions between out-of-plane and in-plane modes of B atoms contribute mainly to the high-frequency region from 500 to 920 cm^{−1}. The low-frequency phonons have a large contribution (81%) to the total EPC (λ = 0.43). Similarly, relatively large values of λ_{qν} along the Γ–M–X directions are found, but they are obviously smaller than that of tetr-W_{2}B_{2}. This explains why its T_{c} is that small. As with our previous study of β_{0}-PC,^{78} the low-frequency phonons in tetr- and hex-W_{2}B_{2} are key to achieving their EPC. Overall, we demonstrate that they are both weak conventional superconductors with EPC constants λ < 1. We also explore the sensitivity of T_{c} with the pseudopotential parameter μ* being 0.08–0.15. As shown in Fig. 7, one can clearly see that there is a decrease of T_{c} with increasing μ* for our studied superconductors.
In the analysis of the electron–phonon interaction in materials containing heavy elements, SOC is generally needed. Since our hypothetical structures contain the same atoms, we only calculate the superconducting properties of tetr-W_{2}B_{2} with SOC as an example in order to save computing resources. When considering SOC, the soft modes along Γ–M harden (Fig. 8), which leads to a decrease of EPC and thus a small value of T_{c}. The superconductive parameters of λ = 0.56, ω_{log} = 268.13 K, and T_{c} = 4.8 K are obtained with the inclusion of SOC for tetr-W_{2}B_{2}. Comparing with the results (λ = 0.69, ω_{log} = 232.4 K, and T_{c} = 7.8 K) without SOC, the superconducting transition temperature decreases by 38.5%. We could deduce that the T_{c} of hex-W_{2}B_{2} would be equal to 0.92 K with the inclusion of SOC.
Fig. 8 (a) Phonon dispersions and (b) the magnitude of the EPC λ_{qν} without/with SOC for tetr-W_{2}B_{2}. The magnitude of λ_{qν} is displayed with the identical scale for comparison. |
Actually, superconductivity has little been predicted for intrinsic 2D systems. In Table 2, we list some analogous 2D phonon-mediated intrinsic superconductors for comparison. These systems have been predicted to exhibit superconductivity without external conditions of high pressure, strain, carrier doping, metal decorations/intercalations, and/or functional groups etc. We can see that the T_{c} of tetr-W_{2}B_{2} is larger than that of B (α sheet), Li_{2}B_{7}, tetr-Mo_{2}B_{2}, tri-Mo_{2}B_{2}, and hex-W_{2}B_{2} while smaller than that of B (β_{12}), borophene, and B_{2}C. Comparing with the intrinsic B monolayer (β_{12} and borophene), the 2D systems of W_{2}B_{2} constrain the vibrations of B atoms and result in smaller values of T_{c}, especially for hex-W_{2}B_{2}. This phenomenon has also been observed in the systems of tetr- and tri-Mo_{2}B_{2}.^{74}
Compounds | μ* | N(E_{F}) | ω _{log} | λ | T _{c} | Ref. |
---|---|---|---|---|---|---|
B (β_{12}) | 0.1–0.15 | 8.12 | 425 | 0.69 | 14 | 79 |
B (α sheet) | 0.05 | 5.85 | 262.2 | 0.52 | 6.7 | 80 |
Borophene | 0.1 | 421.3 | 0.79 | 19 | 81 | |
B_{2}C | 0.1 | 314.8 | 0.92 | 19.2 | 82 | |
Li_{2}B_{7} | 0.12 | 462.8 | 0.56 | 6.2 | 83 | |
tetr-Mo_{2}B_{2} | 0.1 | 16.02 | 344.84 | 0.49 | 3.9 | 74 |
tri-Mo_{2}B_{2} | 0.1 | 16.81 | 295.0 | 0.3 | 0.2 | 74 |
tetr-W_{2}B_{2} | 0.1 | 12.46 | 232.4 | 0.69 | 7.8 | This work |
hex-W_{2}B_{2} | 0.1 | 13.60 | 232.2 | 0.43 | 1.5 | This work |
Charge-carrier doping can generally be applied to control the electronic properties as well as the superconductivity of 2D systems and has been successfully realized.^{78,79,84,85} Here, we also want to investigate the effects of carrier doping on the superconductivity of tetr- and hex-W_{2}B_{2}. We simulate the carrier doping by directly adding electrons into or removing electrons from the systems, together with a compensating uniform charge background of opposite sign to maintain the charge neutrality.^{79,86} For each doping concentration, we relax the plane lattice constants and atomic coordinates. The phonon spectra at different doping concentrations, shown in Fig. 9, indicate a dynamically stable nature, while beyond these concentrations, these two systems are unstable. For tetr-W_{2}B_{2}, we apply both hole doping and electron doping ranging from 0.2 h per cell to 0.2 e per cell. For easy comparison, we only plot the phonon spectra under carrier doping of 0.2 h per cell, 0.1 and 0.2 e per cell [see Fig. 9(a)]. We can see that from 0.2 h per cell to 0.1 e per cell, the phonon spectra are wholly softened, leading to a larger EPC. However, when the doping level gets to 0.2 e per cell, the phonon spectra rebound a little, leading to a decrease of the λ as well as the T_{c}. Thus, the largest value of T_{c} = 9.5 K appears at the electron doping of 0.1 e per cell [see Fig. 9(b)]. Interestingly, we find that the T_{c} values of tetr-W_{2}B_{2} and phosphorene are both about 9.5 K when x = 0.1 e per cell.^{87} With THE inclusion of SOC, the T_{c} of tetr-W_{2}B_{2} should be 5.84 K when x = 0.1 e per cell. As for hex-W_{2}B_{2}, the doping range is from 0.2 h per cell to 0.1 e per cell [see Fig. 9(c)]. Here, we only plot the phonon spectra at the doping concentrations of 0.2 h per cell and 0.1 e per cell. It is clear that hole doping can harden the phonons while electron doping can soften them. From 0.2 h per cell to 0.1 e per cell [see Fig. 9(d)], the EPC as well as the T_{c} gradually increased, obtaining the highest T_{c} of 2.6 K (x = 0.1 e per cell). The highest T_{c} of hex-W_{2}B_{2} reduces by about 38.5% to 1.60 K with SOC. Anyway, as with many other 2D systems, carrier doping can also regulate the superconductivity of our present systems.
As monolayer PtSe_{2}^{88} and borophenes^{68,69} have been grown on a Pt(111) substrate and Ag(111) surface, respectively, our predicted 2D systems of tetr- and hex-W_{2}B_{2} may also be grown on suitable substrates. Since different substrates would induce different strains on them, we explore the superconductivity of tetr- and hex-W_{2}B_{2} under different strains. For each biaxial strain (ξ), calculated by (a positive value means tensile strain while a negative one means compressive strain), atomic coordinates are fully relaxed. We find that tetr-W_{2}B_{2} has no imaginary frequency in the strain range of −2% < ξ < 3%. For easy analysis, we only plot the phonon spectra under the strain of −2%, 0% and 3% in Fig. 10(a). It is clear that the low-energy phonon modes are hardened from −2% to 3%. According to the former discussion, the most significant contributions to the EPC are from these low-energy phonon modes. As a result, the highest T_{c} = 15.4 K with λ = 1.9 is obtained under 2% compressive strain, where some soft modes appear around the M point. The highest T_{c} under 2% compressive strain should be 9.47 K with SOC. On the contrary, for hex-W_{2}B_{2}, the phonon spectra are softened from −1% to 4% [see Fig. 10(c)]. λ and T_{c} increase almost monotonically. Under a tensile strain of 4%, the phonon spectra are fundamentally softened to show soft modes near the M point and along the Γ–K direction, enhancing the T_{c} to 5.9 K with λ = 0.80. With SOC, the enhanced T_{c} is 3.62 K under a tensile strain of 4%. Generally speaking, compressive strains can enhance the EPC of tetr-W_{2}B_{2} while tensile strains can enhance that of hex-W_{2}B_{2}.
The stability and mechanical properties are fundamental aspects for 2D materials in practical applications, especially for novel 2D materials. Although we have shown the dynamical stability of tetr- and hex-W_{2}B_{2} with different carrier doping and biaxial strain [see Fig. 9(a) and (c) and 10(a) and (c)], we still want to study their thermal stability at elevated temperatures, such as room temperature or even higher. We perform 10 ps long AIMD simulations at a series of different temperatures (300, 900, 1500, and 2100 K) and we adopt a 3 × 3 × 1 supercell to minimize the effects of periodic boundary conditions and to explore possible structural reconstruction. The structures after 10 ps at each temperature are shown in Fig. 11. We find that the structure of tetr-W_{2}B_{2} is able to maintain structural integrity up to a rather high temperature of 2100 K. Meanwhile, the structure of hex-W_{2}B_{2} breaks down at a temperature of 900 K. This high thermal stability for tetr-W_{2}B_{2} will make it easy to fabricate in the future. This advantage is favorable for applications in nanoelectronics under ambient conditions and at high temperature.
Fig. 11 Top and side views of structures of (a–d) tetr- and (e–g) hex-W_{2}B_{2} after 10 ps of simulated annealing at different temperatures. |
To check the mechanical stabilities of tetr- and hex-W_{2}B_{2}, we calculate the elastic constants C_{ij} based on the following formula
(6) |
Compounds | C _{11} | C _{12} | C _{22} | C _{66} | Y_{x} | Y_{x} | ν _{ x } | ν _{ y } |
---|---|---|---|---|---|---|---|---|
tetr-W_{2}B_{2} | 245 | 151 | 245 | 156 | 152 | 152 | 0.62 | 0.62 |
hex-W_{2}B_{2} | 284 | 61 | 299 | 109 | 271 | 286 | 0.20 | 0.22 |
Recently, following the directions given by first-principle simulations, borophene (boron monolayer) has been successfully grown on a Ag(111) surface under an ultrahigh vacuum.^{69,92} Inspired by this, we suggest that tetr- and hex-W_{2}B_{2} should also be directly grown on suitable substrates instead of being exfoliated from their bulk. After extensive investigations, we select the widely used semiconductors of SiC, CuI, and ZnS as ideal candidate substrates for epitaxial growth of tetr- and hex-W_{2}B_{2}. Their detailed structures in the substrate systems are illustrated in Fig. 12. For each calculation, the structures of the substrates are fully relaxed and then their atomic coordinates are fixed to simulate substrates plus W_{2}B_{2}. As a theoretical guide, we explore possible substrates by calculating their adsorption energy and lattice mismatch (δ). The adsorption energy is calculated according to the following equation
(7) |
(8) |
Fig. 12 Top and side views of tetr-W_{2}B_{2} on (a) SiC(100), (b) CuI(100), and (c) ZnS(100) surfaces and hex-W_{2}B_{2} on (d) SiC(111), (e) CuI(111), and (f) ZnS(111). |
The results of the calculated adsorption energy and lattice mismatch are listed in Table 4. All negative values of adsorption energies indicate that our predicted W_{2}B_{2} monolayers may be grown on such substrates. Besides, a small lattice mismatch is required in the real growing of 2D materials. Considering the small adsorption energy and lattice mismatch of the six promising monolayer–substrate schemes, the growth of tetr-W_{2}B_{2} on ZnS(100) and hex-W_{2}B_{2} on ZnS(111) should be the two most likely cases.
tetr-W_{2}B_{2} | E _{ads} (eV) | δ (%) | hex-W_{2}B_{2} | E _{ads} (eV) | δ (%) |
---|---|---|---|---|---|
SiC(100) | −4.74 | 7.25 | SiC(111) | −1.71 | 0.97 |
CuI(100) | −2.55 | 0.49 | CuI(111) | −2.88 | 7.91 |
ZnS(100) | −4.21 | 2.06 | ZnS(111) | −4.93 | 1.32 |
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