Novel structures of two-dimensional tungsten boride and their superconductivity

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
aKey Laboratory of Key Film Materials & Application for Equipments (Hunan Province), School of Material Sciences and Engineering, Xiangtan University, Xiangtan, Hunan 411105, China. E-mail:;
bHunan Provincial Key Laboratory of Thin Film Materials and Devices, School of Material Sciences and Engineering, Xiangtan University, China
cInstitute of High Energy Physics, Chinese Academy of Sciences (CAS), Beijing 100049, China. E-mail:
dDongguan Institute of Neutron Science (DINS), Dongguan 523808, China
eCollaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, 030006, China
fState Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan, 030006, China
gCollege of Physics and Materials Science, Henan Normal University, Xinxiang 453007, China

Received 14th May 2019 , Accepted 17th June 2019

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-W2B2 and hex-WB4 and investigate their bonding types, electronic properties, phonon dispersions and electron–phonon coupling (EPC). The results show that both tetr- and hex-W2B2 are intrinsic phonon-mediated superconductors with a superconducting transition temperature (Tc) of 7.8 and 1.5 K, respectively, while tri-W2B2 and hex-WB4 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 Tc of tetr-W2B2 can be increased to 15.4 K under a compressive strain of −2%, the Tc of hex-W2B2 can be enhanced to 5.9 K by a tensile strain of +4%. With the inclusion of spin–orbit couping (SOC), the value of Tc decreases by 38.5% in our systems. Furthermore, we explore the stabilities and mechanical properties of tetr- and hex-W2B2 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.

1 Introduction

Superconductivity is a unique quantum phenomenon characterized by zero electrical resistance and the Meissner effect. It arises due to the Cooper pair formation, even in two-dimensional (2D) systems.1 2D superconductors have many interesting properties, such as localization of electrons and/or Cooper pairs,2 transition-temperature oscillations caused by quantum size effects,3–5 excess conductivity originating from superconducting fluctuations,6–8 Berezinskii–Kosterlitz–Thouless transitions9,10 and quantum phase transitions at zero temperature. However, 2D superconductivity is on the verge of a transition to a metallic or an insulating state and thus could be fragile. It is usually lost when the thickness of a metal film approaches 1–2 nm.11 Therefore, there is a need to fabricate 2D superconductors in highly ordered and controlled systems and probe their superconducting properties using advanced techniques.12

In recent years, rapid advancements in nanotechnology, including molecular beam epitaxy,13 atomic layer deposition,14 pulsed laser deposition,15in situ ultrahigh vacuum-low temperature measurements,16 scanning tunneling microscopy/spectroscopy17etc., have greatly improved the studies on ultrathin 2D superconductors. For example, the interface between LaAlO3 and SrTiO3, 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 SrTiO3 and ZrNCl.19,20 For a 1 unit cell thick FeSe layer, epitaxially grown on SrTiO3 substrates, its Tc 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-intercalation25,26 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 conditions34 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-W2B2 and hex-WB4 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-W2B2 are intrinsic superconductors with Tc of 7.8 and 1.5 K, respectively, while tri-W2B2 and hex-WB4 don't exhibit superconducting properties. The value of Tc 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-W2B2 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-W2B2, respectively.

image file: c9cp02727k-f1.tif
Fig. 1 Top (left panel) and side views (middle panel) as well as corresponding 2D BZs (right panel) for 2D (a–c) tetr-W2B2, (d–f) hex-W2B2, (g–i) tri-W2B2, and (j–l) hex-WB4. The Mo and B atoms are denoted by red and blue spheres, respectively.

2 Computational methods

We search the crystal structures of 2D TBs by utilizing the swarm intelligence-based CALYPSO structure prediction method.49,50 The structural optimizations are performed at the density-functional theory (DFT) level as implemented in the Vienna ab initio simulation package (VASP).51 The exchange–correlation functional is treated within the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE) type, which has widely been used in exploring new materials and new structures.52,53 The electron–ion interactions are described by the projector-augmented-wave potentials with 5d46s2 and 2s22p1 configurations treated as valence electrons for W and B, respectively. A kinetic cutoff energy of 550 eV and 10 × 10 × 1 Monkhorst–Pack (MP) k-point mesh are adopted in our calculations. To avoid the nonphysical coupling between adjacent sheets along the out-plane c axis, we set a vacuum thickness of 15 Å. We also use a rapid dispersion-corrected DFT method (optB86b-vdW)54–56 to describe the van der Waals interaction. Geometries are fully relaxed until the residual forces on each atom become less than 0.01 eV Å−1. To test the thermal stabilities of our predicted stable and meta-stable structures, we perform AIMD simulations at a series of temperatures (300, 900, 1500 and 2100 K). All these simulations, using the constant number, volume, and temperature (NVT) ensemble, last for 10 ps with a time step of 1 fs. A 3 × 3 × 1 supercell is adopted. The temperature is controlled using the Nosé–Hoover thermostat.57

The electronic structures and superconducting properties are calculated by using plane-wave code QUANTUM ESPRESSO (QE).58 The norm-conserving GGA pseudopotentials59 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-W2B2 and hex-WB4 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-W2B2 and 4 × 4 × 1 for tri-W2B2 and hex-WB4. The phonon modes and EPC are calculated using the density-functional perturbation theory (DFPT)60 and the Eliashberg equation.61,62 The Tc is evaluated by using the Allen–Dynes modified McMillan formula63 with a typical Coulomb pseudopotential of μ* = 0.1.64–66

3 Results and discussion

Using CALYPSO together with VASP, thousands of new structures of 2D TBs are generated after structure-searching calculations and queued in order of enthalpy from low to high. For so many structures, one may mainly focus on the one with the global minimum energy. However, some low-lying or even relatively high energy allotropes may also be synthesized under suitable conditions.67 For one typical example, several borophene allotropes with relatively high energy have been fabricated experimentally.68–70 Inspired by this, we select four structures of 2D TBs by performing accurate optimizations and phonon spectrum calculations, three for W2B2 and one for WB4. For simplicity, we label them as tetr-, hex-, and tri-W2B2 and hex-WB4, respectively, according to their different space groups. We present in Fig. 1 their crystal structures along with the corresponding 2D BZs. tetr-W2B2 crystallizes in a tetragonal phase with a space group of P4/nmm (no. 129). This structure can be viewed as a distorted NaCl-type structure like MX compounds (X = C and N and M = Sc, Ti, V, Cr, Mn, Fe, Co, and Ni).71 W and B atoms connect with each other in the in-plane and out-of-plane layers. tetr-W2B2 has the minimum energy among the three types of 2D W2B2. hex-W2B2 crystallizes in a hexagonal phase with a space group of P6/mmm (no. 191). This is a layered structure with a hexagonal B atomic layer surrounded by two trigonal W layers. The formation energy of this metastable structure is higher by 266 meV per atom than that of the global minimum one. The obtained tri-W2B2 belongs to the trigonal P[3 with combining macron]m1 (no. 164) space group, stacking in order of W–B–W in the vertical direction. Its distorted boron honeycomb sheet is in between two hexagonal planes of W. This metastable structure is energetically higher by 107 meV per atom than that of tetr-W2B2. hex-WB4 crystallizes in a hexagonal phase with a space group of P6/mmm (no. 191). It also consists of three sublayers stacked in the order of B–W–B. The two B sublayers are strongly bonded with the middle W plane. The high-symmetry paths, used for our following plotting of the band structures and phonon spectra, of all these structures are clearly shown in the 2D BZs.

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 (QB), bond lengths, and charge density at the corresponding bond point (CDb) are listed in Table 1. As a general rule, when the value of CDb 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 CDb 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.

image file: c9cp02727k-f2.tif
Fig. 2 Top and side views of the charge density (top panels) and difference in charge density (bottom panels) for (a) tetr-W2B2, (b) hex-W2B2, (c) tir-W2B2 and (d) hex-WB4. The yellow and cyan areas represent electron gains and losses, respectively.
Table 1 Bader charge (QB), bond lengths, and charge density at the corresponding bond point (CDb) for tetr-, hex-, and tri-W2B2 and hex-WB4. In tetr-W2B2, there are two typical kinds of W–B bonding: one is within the xy plane and another along the c direction. The bonding along the c direction is indicated by the data within parentheses
Compounds Q B (W) Q B (B) B–B W–W W–B CDb (B–B) CDb (W–W) CDb (W–B)
tetr-W2B2 5.16 3.84 3.07 3.07 2.19(2.62) 0.046 0.045 0.081(0.070)
hex-W2B2 5.49 3.51 1.76 3.05 2.28 0.113 0.043 0.073
tri-W2B2 5.67 3.33 1.75 2.85 2.18 0.118 0.050 0.064
hex-WB4 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-W2B2: (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 CDb (W–B) = 0.070 (0.081) e a.u.−3.

For hex-W2B2, B atoms connect with each other by strong covalent bonds within the B atomic layer with CDb (B–B) = 0.113 e a.u.−3 while the bonding between B and W layers is relatively weak. As for tri-W2B2, its bonding features are quite similar to that of hex-W2B2, but with shorter W–B and W–W bonds. This answers to some extent why tri-W2B2 is more stable. Considering the strong covalent bonds with CDb (B–B) = 0.118 e a.u.−3 in tri-W2B2, 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-W2B2 while there are only one B–B bond and four W–B bonds in tri-W2B2. Thus, the fact that tetr-W2B2 is more stable than tri-W2B2 is understandable. For hex-WB4, 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-W20.84−B20.84+, hex-W20.51−B20.51+, tri-W20.33−B20.33+ and hex-W0.31−B40.31+, respectively. It is easy to conclude that the electron transfer of tetr-W2B2 is the most dominant among all our predicted 2D TBs, even more dominant than that of tetr-Mo2B2, 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-W2B2 are the strongest, consistent with our previous charge density analyses. As for hex-W2B2, tri-W2B2, and hex-WB4, 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)].

image file: c9cp02727k-f3.tif
Fig. 3 ELF planes of (a) tetr-, (b) hex-, and (c) tri-W2B2 and (d) hex-WB4 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-W2B2, 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-W2B2, as presented in Fig. 4(d) and 5(c), there are three and four bands crossing the Fermi level along ΓM and ΓK, 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-W2B2, 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-WB4, there are one and three crossings along the MK and ΓK paths, respectively [see Fig. 4(j)]. These crossings form two Dirac cones, one at K and another half way along ΓK, near the Fermi energy level. The inclusion of SOC leads to the degeneration of bands in the whole BZ for 2D W2B2. For hex-W2B2 and hex-WB4 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-W2B2 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 W2B2, 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-WB4, 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 W2B2 structures is mainly controlled by their W-5d orbitals while that of hex-WB4 is governed by the B-2p orbitals.

image file: c9cp02727k-f4.tif
Fig. 4 Orbital-resolved band structures without SOC (left one) and with SOC (right one) as well as total and partial density of sates calculated without SOC (middle one) of (a–c) tetr-, (d–f) hex-, (h and g) and tri-W2B2 and (k–m) hex-WB4. The Fermi energy level is set as zero.

image file: c9cp02727k-f5.tif
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-W2B2 and (h and i) hex-WB4.

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-W2B2, the highest phonon frequency is 920 cm−1, which is the highest in our obtained TBs and also higher than that of Mo2B2 (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-W2B2.

image file: c9cp02727k-f6.tif
Fig. 6 Phonon dispersions, PhDOS, Eliashberg spectral function α2F(ω), and cumulative frequency dependence of EPC λ(ω) of (a–d) tetr-W2B2 and (e–h) hex-W2B2. The phonon dispersions are weighted by the motion modes of W and B atoms as well as the magnitude of EPC λimage file: c9cp02727k-t1.tif in the first-left and the second-left panels, respectively. (i and k) The phonon dispersions and (j and l) PhDOS of tri-W2B2 and hex-WB4. The red, pink, blue and orange hollow circles in (a, e, i and k) indicate W horizontal, W vertical, B horizontal and B vertical modes, respectively. The magnitude of λ is displayed using an identical scale in all figures for comparison.

The phonon dispersions can be weighted by the magnitude of the EPC λ. According to the Migdal–Eliashberg theory,77 the λ is calculated by

image file: c9cp02727k-t2.tif(1)
where γ is the phonon linewidth, ω is the phonon frequency and N(EF) is the electronic density of states at the Fermi level. The Eliashberg spectral function α2F(ω) is estimated based on the Eliashberg equation:61
image file: c9cp02727k-t3.tif(2)
The total EPC constant is calculated by the frequency-space integration:
image file: c9cp02727k-t4.tif(3)
and the logarithmic average frequency ωlog is calculated by
image file: c9cp02727k-t5.tif(4)
Based on BCS theory46 and the above results, the superconducting transition temperature Tc can be calculated by using the McMillian–Allen–Dynes formula:
image file: c9cp02727k-t6.tif(5)
where μ* is the effective screened Coulomb repulsion constant with a typical value of μ* = 0.1. The results of the EPC λ, phonon density of states (PhDOS), Eliashberg spectral function α2F(ω), and the cumulative frequency dependence of EPC λ(ω) are exhibited in Fig. 6(b)–(d) and (f)–(h) for tetr- and hex-W2B2, respectively. We demonstrate that they are intrinsic superconductors with Tc being 7.8 and 1.5 K, respectively. However, according to our calculated EPC, tri-W2B2 and hex-WB4 are not superconductors. Next, we will only focus on tetr- and hex-W2B2.

From our calculated cumulative frequency dependence of EPC λ(ω) for tetr-W2B2, 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 along the ΓMX directions in the frequency range of 65–130 cm−1 result in two large peaks on the PhDOS and α2F(ω). As a consequence, λ(ω) increases rapidly in this frequency range and the large values of λ here are the main origin of its superconductivity.

As for hex-W2B2, the W atomic vibrations (Wz and Wxy 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 along the ΓMX directions are found, but they are obviously smaller than that of tetr-W2B2. This explains why its Tc is that small. As with our previous study of β0-PC,78 the low-frequency phonons in tetr- and hex-W2B2 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 Tc 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 Tc with increasing μ* for our studied superconductors.

image file: c9cp02727k-f7.tif
Fig. 7 Calculated superconducting critical temperature Tc as a function of Coulomb pseudopotential parameter μ*. The red line represents tetr-W2B2 and the blue line represents hex-W2B2.

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-W2B2 with SOC as an example in order to save computing resources. When considering SOC, the soft modes along ΓM harden (Fig. 8), which leads to a decrease of EPC and thus a small value of Tc. The superconductive parameters of λ = 0.56, ωlog = 268.13 K, and Tc = 4.8 K are obtained with the inclusion of SOC for tetr-W2B2. Comparing with the results (λ = 0.69, ωlog = 232.4 K, and Tc = 7.8 K) without SOC, the superconducting transition temperature decreases by 38.5%. We could deduce that the Tc of hex-W2B2 would be equal to 0.92 K with the inclusion of SOC.

image file: c9cp02727k-f8.tif
Fig. 8 (a) Phonon dispersions and (b) the magnitude of the EPC λ without/with SOC for tetr-W2B2. The magnitude of λ 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 Tc of tetr-W2B2 is larger than that of B (α sheet), Li2B7, tetr-Mo2B2, tri-Mo2B2, and hex-W2B2 while smaller than that of B (β12), borophene, and B2C. Comparing with the intrinsic B monolayer (β12 and borophene), the 2D systems of W2B2 constrain the vibrations of B atoms and result in smaller values of Tc, especially for hex-W2B2. This phenomenon has also been observed in the systems of tetr- and tri-Mo2B2.74

Table 2 Superconductive parameters of μ*, N(EF) (in units of states/spin/Ry/cell), ωlog (in K), λ and Tc (in K) for some intrinsic 2D phonon-mediated superconductors
Compounds μ* N(EF) ω log λ T c Ref.
B (β12) 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
B2C 0.1 314.8 0.92 19.2 82
Li2B7 0.12 462.8 0.56 6.2 83
tetr-Mo2B2 0.1 16.02 344.84 0.49 3.9 74
tri-Mo2B2 0.1 16.81 295.0 0.3 0.2 74
tetr-W2B2 0.1 12.46 232.4 0.69 7.8 This work
hex-W2B2 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-W2B2. 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-W2B2, 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 Tc. Thus, the largest value of Tc = 9.5 K appears at the electron doping of 0.1 e per cell [see Fig. 9(b)]. Interestingly, we find that the Tc values of tetr-W2B2 and phosphorene are both about 9.5 K when x = 0.1 e per cell.87 With THE inclusion of SOC, the Tc of tetr-W2B2 should be 5.84 K when x = 0.1 e per cell. As for hex-W2B2, 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 Tc gradually increased, obtaining the highest Tc of 2.6 K (x = 0.1 e per cell). The highest Tc of hex-W2B2 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.

image file: c9cp02727k-f9.tif
Fig. 9 Phonon spectra with different carrier doping for (a) tetr- and (c) hex-W2B2. Variations of λ and Tc along with different carrier doping for (b) tetr- and (d) hex-W2B2. The red and blue lines correspond to the Tc and λ, respectively.

As monolayer PtSe288 and borophenes68,69 have been grown on a Pt(111) substrate and Ag(111) surface, respectively, our predicted 2D systems of tetr- and hex-W2B2 may also be grown on suitable substrates. Since different substrates would induce different strains on them, we explore the superconductivity of tetr- and hex-W2B2 under different strains. For each biaxial strain (ξ), calculated by image file: c9cp02727k-t7.tif (a positive value means tensile strain while a negative one means compressive strain), atomic coordinates are fully relaxed. We find that tetr-W2B2 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 Tc = 15.4 K with λ = 1.9 is obtained under 2% compressive strain, where some soft modes appear around the M point. The highest Tc under 2% compressive strain should be 9.47 K with SOC. On the contrary, for hex-W2B2, the phonon spectra are softened from −1% to 4% [see Fig. 10(c)]. λ and Tc 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 ΓK direction, enhancing the Tc to 5.9 K with λ = 0.80. With SOC, the enhanced Tc is 3.62 K under a tensile strain of 4%. Generally speaking, compressive strains can enhance the EPC of tetr-W2B2 while tensile strains can enhance that of hex-W2B2.

image file: c9cp02727k-f10.tif
Fig. 10 Phonon spectra under different biaxial strains for (a) tetr- and (c) hex-W2B2. Variations of λ and Tc under biaxial strain for (b) tetr- and (d) hex-W2B2. The red and blue lines correspond to the Tc and λ, respectively.

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-W2B2 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-W2B2 is able to maintain structural integrity up to a rather high temperature of 2100 K. Meanwhile, the structure of hex-W2B2 breaks down at a temperature of 900 K. This high thermal stability for tetr-W2B2 will make it easy to fabricate in the future. This advantage is favorable for applications in nanoelectronics under ambient conditions and at high temperature.

image file: c9cp02727k-f11.tif
Fig. 11 Top and side views of structures of (a–d) tetr- and (e–g) hex-W2B2 after 10 ps of simulated annealing at different temperatures.

To check the mechanical stabilities of tetr- and hex-W2B2, we calculate the elastic constants Cij based on the following formula

image file: c9cp02727k-t8.tif(6)
where Es is strain energy and the tensile strain is defined as image file: c9cp02727k-t9.tif, and a and a0 are the lattice constants of the strained and strain-free structures, respectively. In order to calculate the elastic stiffness constants, Es as a function of ε at strains of −2% ≤ ε ≤ 2% with an increment of 0.5% is calculated. Then, the Cij can be obtained by postprocessing the VASP calculated data using the VASPKIT code.89 On the basis of the calculated Cij, the in-plane Young's modulus (Y) of this material along the x and y directions can be calculated by image file: c9cp02727k-t10.tif and image file: c9cp02727k-t11.tif, respectively. The Poisson's ratio can be calculated by image file: c9cp02727k-t12.tif and image file: c9cp02727k-t13.tif. The results of Cij, Young's modulus and Poisson's ratio for tetr and hex-W2B2 are listed in Table 3. Obviously, according to the Born criteria90 of a mechanically stable 2D structure: C11C12C122 > 0 and C66 > 0, they are mechanically stable. The values of Young's modulus are much larger than that of MoS2 (123 N m−1)91 and the Poisson's ratios are smaller than that of 2D WB2 (0.67),36 showing outstanding mechanical properties. This means that they would only shrink slightly in the x direction when stretched in the y direction.

Table 3 Calculated parameters of Cij (N m−1), Young's modulus (N m−1) and Poisson's ratio for tetr- and hex-W2B2
Compounds C 11 C 12 C 22 C 66 Yx Yx ν x ν y
tetr-W2B2 245 151 245 156 152 152 0.62 0.62
hex-W2B2 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-W2B2 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-W2B2. 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 W2B2. 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

image file: c9cp02727k-t14.tif(7)
where Emol+slab denotes the total ground state energy of the optimized configuration of 2D W2B2 adsorbed on the slab surface, Eslab is the total energy of the slab with a clean surface, Emol is the energy of 2D W2B2 and n is the 2D W2B2 atom number. The lattice mismatch (δ) is calculated using
image file: c9cp02727k-t15.tif(8)
where Lmol and Lslab are the corresponding lattice parameters of 2D W2B2 and the slab, respectively.

image file: c9cp02727k-f12.tif
Fig. 12 Top and side views of tetr-W2B2 on (a) SiC(100), (b) CuI(100), and (c) ZnS(100) surfaces and hex-W2B2 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 W2B2 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-W2B2 on ZnS(100) and hex-W2B2 on ZnS(111) should be the two most likely cases.

Table 4 Adsorption energies and the lattice mismatch (δ) for tetr- and hex-W2B2 on various substrates
tetr-W2B2 E ads (eV) δ (%) hex-W2B2 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

4 Conclusion

In summary, by combining the crystal structure prediction technique and the first-principles method, we have obtained four novel structures of 2D TBs and studied their bond types, electronic properties and even superconductivity. Surprisingly, we have found that tetr- and hex-W2B2 are intrinsic superconductors with a Tc of 7.8 and 1.5 K, respectively, while tri-W2B2 and hex-WB4 are normal metals. In order to study these two new 2D superconductors deeply, we have further investigated the effects of carrier doping and biaxial strain. The results show that these external conditions can soften the low-frequency phonon modes and enhance the EPC strength, leading to an increase in the Tc. Furthermore, we have also explored the stabilities and mechanical properties of tetr- and hex-W2B2. While considering SOC, the value of Tc decreases by 38.5% for the superconductors. Significantly, their outstanding thermal stabilities and mechanical properties indicate that they could be applied under harsh conditions. In order to provide clues for the synthesis of these new 2D superconductors, we have recommended that they may be prepared by growing on ZnS(100) and ZnS(111) substrates, respectively. We believe that our findings will broaden the novel 2D superconducting family and would inspire further efforts in this field.

Conflicts of interest

There are no conflicts to declare.


The authors would like to acknowledge the financial support from the National Natural Science Foundation of China under Grant No. 11835008 and 51872250, the State Key Laboratory of Intense Pulsed Radiation Simulation and Effect (Northwest Institute of Nuclear Technology) under Grant No. SKLIPR1814, and the Key Laboratory of Low Dimensional Materials & Application Technology under the Ministry of Education (Xiangtan University) under Grant No. KF20180203. P. F. L. and B. T. W. also acknowledge the PhD Start-up Fund of Natural Science Foundation of Guangdong Province of China (No. 2018A0303100013) and the Program of State Key Laboratory of Quantum Optics and Quantum Optics Devices (No. KF201904). T. B. acknowledges the China Postdoctoral Science Foundation under Grant No. 2018M641477. The calculations were performed at the Supercomputer Centre at the China Spallation Neutron Source.

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