3. Results and Discussions
First, the structural features of the graphdiyne monolayers explored in this study are discussed.
Figure 1 exhibits the primitive cells for four variants of graphdiyne nanosheets, made of structurally similar core molecules of benzotrithiophene (BTT: C
12S
3H
6), benzotripyrrol (BTP: C
12N
3H
9), benzotrifuran (BTF: C
12O
3H
6) and benzotri (1H-Pyrrol-1-yl) (BTHP: C
12N
3H
6) with C
3h symmetry, which are covalently held together by diacetylene linkers. For a consistent reference to these 2D polymers, they are called BTX-GDYs, where X stand for T, P, F and PH. BTX-GDYs are topologically flat lattices, and each hexagonal primitive cell includes two core molecules and three diacetylene linkers. The optimized lattice constants are close to each other, and vary slightly from 22.82 Å for the BTHP-GDY to 23.67 Å for the BTT-GDY monolayer. The predicted lattice constant for the BTT-GDY lattice is considerably close but is still around 2% larger than the experimental data for the multilayer BTT-GDY (a = b = 23.2 Å) [
19]. This can be attributed to the fact that under experimental conditions, due to existence of large porosity and thermal effects, the structures do not stay fully flat and contract, resulting in a slightly lower lattice constant. The structural and electronic properties of the BTX-GDYs monolayers are summarized in
Table 1, and complete atomic structures are included in the
Supplementary Materials. According to the measured bond lengths summarized in
Table 1, as expected, covalent interactions are dominant within the planes of the BTX-GDYs nanosheets. For a bird’s eye view of chemical interactions within the in-plane and over-the-plane of BTX-GDYs, we calculated the electron localization function (ELF) [
33], which is illustrated in
Figure 1 and
Figure S1, respectively. In
Figure 1, the localization pattern (alternating toric and rectangular red cross sections) for alternating between single and triple C-C bonds in diacetylene linkers is also conspicuous. The localization pattern in between those carbon atoms binds BTX and diacetylene linkers, which indicates a charge transfer from the former to the latter. From the Bader charge analysis, we found that BTX molecules in BTPH-, BTP-, BTF- and BTT-GDY transfer on average 0.47, 0.24, 0.38 and 0.44
e to each diacetylene linker, respectively. In addition, the iso-surface ELF maps of BTX-GDYs plotted in
Figure S1 highlight that polymerization enables efficient delocalization of the π system of the BTX molecules through diacetylene linkers. This is expected to lead to smaller band gap values for the BTX-GDYs with respect to the HOMO-LUMO gaps of the BTX molecules, and also promote the charge transport within the planes.
After analyzing the structural features of the BTX-GDY nanosheets, we next more elaborately investigated their electronic features. To this goal, electronic band structure calculations were carried out using the standard PBE and more accurate HSE06 functional. For each lattice, the HSE06 electronic band structure, atom-type projected density of states (pDOS), atomic orbital pDOS, contribution of diacetylene linkers to the DOS and charge density distributions plots for the valence band maximum (VBM) and conduction band minimum (CBM) are illustrated in
Figure 2. The corresponding electronic band structures with PBE are included in
Figure S2, and band gap values are also summarized in
Table 1. According to the presented results, the BTT-GDY monolayer is a direct gap semiconductor at Γ point with an estimated HSE06 (PBE) band gap of 2.53 (1.88) eV. The predicted band gap by HSE06 is 0.15 eV larger than the experimental data (2.38 eV) for the multilayer BTT-GDY [
19], which is remarkably close and might be an indication of a limited quantum confinement effect in this system. The band gap of the BTT-GDY monolayer is also appreciably larger than that of the γ-graphdiyne (0.89 eV) [
34]. In the BTT-GDY monolayer, both valence and conduction bands (VB, CB) are almost flat and parallel, indicating very large effective masses (62.5 m
0 and 5.26 m
0 for hole and electrons, respectively) and expectably low carrier mobilities. The VBM at the Γ point is, however, two-fold degenerate, and the second VB is highly dispersed along Γ→M and Γ→K, suggesting small effective masses (0.37
m0) and possibly large hole mobilities. One can clearly observe from
Figure 2 that the BTF- and BTP-GDYs show qualitatively similar band structures to the BTT-GDY counterpart, with the minor difference that the CB of the BTP-GDY is more dispersed than that of the other two counterparts. The band gaps of BTF-GDY and BTP-GDY with HSE06 (PBE) were found to be 2.65 (1.94) eV and 2.49 (1.83) eV, respectively, almost insensitive to the chemistry of core molecules. The calculated band gaps are appreciably smaller than the HOMO-LUMO gaps (E
H-L) of the BTX and diacetylene molecules, as listed in
Table 1. The absolute band edge positions (E
VBM, E
CBM) are also observed to straddle the HOMO and LUMO energy levels (E
H, E
L) of BTXs and diacetylene molecules (
Table 1). For each 2D polymer, the VBM lies energetically close to the HOMO level of the BTX, while the CBM is located appreciably lower than the LUMO level of the BTX (see
Table 1). The band structure analysis presented in
Figure 2 indicates that for the aforementioned three cases, both the VBM and CBM exhibit a π character made of p
z orbitals of C and hetero-atoms (N, O, S), which are distributed over both the BTX and, at the minimum, over two diacetylene linkers. The VBM can be thought of as a non-bonding interaction between HOMOs of the BTX molecule and the diacetylene linker, while the CBM represents an obvious bonding hybridization between a π state of the BTX molecule and LUMO of diacetylene linkers (shown with dashed red rectangles in charge density distributions plots in
Figure S3). This finding explains the underlying mechanism for why almost the same energy levels for the VBM of the BTX-GDY monolayers and HOMO of the BTX molecule and much deeper energy levels for the CBM of BTX-GDY monolayers were observed, as compared to the LUMO of the BTX molecule. To further support the aforementioned discussion, charge density distribution of HOMO and LUMO states of the BTX and diacetylene molecules is shown in
Figure S4. In spite of having the same number of electrons per primitive cell and almost similar electronic structures, the BTP-GDY monolayer exhibits a much smaller mid-gap work function (3.64 eV) than those of BTT- and BTF-GDYs, at 4.35 and 4.30 eV, respectively. Unlike these three, BTHP-GDY includes six less electrons. The BTHP-GDY system can be simply generated by removing six hydrogen atoms connected to nitrogen in BTP-GDY’s primitive cell. Taking into account that the bands around the Fermi level in BTP-GDY show the π character with no contribution from hydrogen atoms, it may not be surprising that the electronic band structures of the BTP- and BTHP-GDYs look almost similar, except for the mid-gap work function of BTHP-GDY, which is much lower, 5.98 eV. By counting down three bands from the Fermi level of BTP-GDY, the band gap of BTHP-GDY occurs between two closely lying bands with an energy difference of 0.06 (0.03) eV. This gap is indirect with the VBM at K and CBM at the Γ point. The predicted narrow band gap for the BTH-GDY nanosheet may limit its practical applications in electronic devices and photocatalytic chemical processes. The other three BTX-GDY nanosheets, however, are appealing candidates due to their sufficiently large band gaps.
After the elaborated investigation of the electronic features of the BTX-GDY monolayers, in the next step, their dynamical stability and mechanical properties will be explored. Before analyzing the phonon dispersion relations, we first examined the accuracy of the developed MLIPs. It is worthwhile to note that the accuracy of MTPs for the evaluation of phonon dispersion relations has been extensively verified in previous studies [
29,
31]. For this purpose, we conducted biaxial tensile straining employing the developed classical models conducted at 1 K and compared the estimations with the ground-state DFT. Because of the large number of atoms and excessive computational costs of DFT simulations, only primitive hexagonal unit cells were considered for these simulations. With the developed classical models, owing to enhanced computational costs, rectangular supercells were considered in these simulations. In the conducted CMD calculations, the atomic displacements along the out-of-plane direction were fixed in order to keep the consistency of the DFT-based results. In order to plot the results in the standard unit of GPa, a consistent thickness of 3.35 Å is was for the BTX-GDY monolayers based on that for the single-layer graphene. In
Figure 3, the stress–strain responses of the BTX-GDY monolayers stretched uniaxially along the armchair and zigzag directions were compared. It is worthwhile to note that since the systems’ dimensions along the perpendicular directions of loading are fixed, this results in the formation of two-axial stress condition in these systems. As can be seen for both loading directions, the developed MTPs can reproduce the stress evolution along the zigzag and armchair directions with an exceptional level of accuracy as compared with the computationally expensive DFT counterpart. In
Figure S5, we also compare the failure mechanism predicted by the DFT and MLIP method for the BTT-GDY monolayer, which confirms consistent predictions by both approaches. Our analysis of the deformation mechanism reveals that upon the uniaxial straining, the failure in the BTX-GDY nanosheets always initiates along the C-C bonds between the core BTX molecule and diacetylene linkers. It is clear that these 2D systems show considerably close mechanical and failure behaviors.
We next investigated the dynamical stability of the BTX-GDY monolayers on the basis of phonon dispersion relation calculated using the developed MTPs. In
Figure 4, the obtained phonon dispersion relations along the high-symmetry directions of the first Brillouin zone are depicted. Due to the presence of H atoms in these 2D polymers, rather flat optical modes appear at frequencies around 95 THz, which are not illustrated. Like graphene and other 2D lattices, the BTX-GDY monolayers also show three acoustic phonon modes appearing at the Γ point. As it can be seen, none of the acoustic and optical modes in these monolayers exhibit imaginary frequencies, confirming their dynamical stability. It is also noticeable that phonon modes in the considered BTX-GDY monolayers show relatively flat dispersions, which reveal their low group velocities. In addition, remarkable band crossing is conspicuous throughout the entire frequency range for both acoustic and optical modes, revealing high scattering and a short phonon lifetime. The combination of low phonon group velocity and high scattering rates suggest remarkably low lattice thermal conductivity along the BTX-GDY monolayers, which might be appealing for designing next-generation thermoelectric energy conversion systems.
Although the uniaxial straining results highlight remarkable strengths of the BTX-GDY nanosheets, in order to provide a more useful vision concerning the mechanical responses, it is critical to conduct uniaxial tensile simulations. After ensuring the accuracy of the accelerated classical models in the analysis of stress–strain relations, one can utilize them to conduct the uniaxial tensile simulations instead of the expensive DFT counterpart. In
Figure 5, the uniaxial stress–strain responses of the BTX-GDY monolayers stretched along the armchair and zigzag directions are compared. On the basis of the developed classical models, the tensile strength of the BTT, BTHP, BTP and BTF-GDY monolayers along the armchair (zigzag) directions was predicted to be 18.1 (14.1), 18.0 (16.2), 18.2 (15.7) and 19.4 (16.2) GPa, respectively. As expected, the predicted values reveal that the mechanical characteristics of the BTX-GDYs show marginal dependency on the chemistry of the core molecules. Nonetheless, for the uniaxial loading along the zigzag direction, the BTT-GDY monolayer clearly shows a lower tensile strength than other lattices. As previously discussed, C-S bonds in the unstrained BTT-GDY monolayer were found to be more than 25% larger than the corresponding C-N and C-O bonds in the other considered lattices, which reveals the softened strength of C-S bonds. As shown in
Figure 5b, for the loading along the zigzag direction, the failure in the BTT-GDY monolayer initiates along the C-S bonds, justifying its lower tensile strength. It is also noticeable that the BTX-GDY nanosheets, instead of conventional linear elasticity, yield nonlinear stress–strain relations at low strain levels. The lack of linear elasticity in these systems reveals that the deformation proceeds with a combination of structural deflection and bond stretching. From the structural point of view, because of the existence of rather long diacetylene linkers, these nanosheets include large porosity in their lattice. As such, during the uniaxial tensile loading, sheets more severely contract along the width than densely packed lattices such as graphene, which facilitates the deformation and results in a smooth stress rise at initial stages of the loading. In contrast with conventional materials, at higher strain levels close to the ultimate tensile strength, the stress starts to increase sharply in the BTX-GDY nanosheets. At high strain levels, due to vdW interactions with neighboring atoms, the contraction along the width becomes limited, and thus the bond stretching starts to become the dominant load carrier, resulting in a sharper stress rise by strain. It is also noticeable that along the armchair direction, the BTX-GDY nanosheets clearly exhibit higher tensile strengths and stretchability than along the zigzag direction, which can be also attributed to the structural features. As previously shown, every unit cell in these systems includes three diacetylene linkers, and their configuration with respect to the loading affects the deformation process. As shown in
Figure 5b, for the loading along the zigzag direction, for every three diacetylene linkers, one stays exactly perpendicular to the loading during the entire loading process. The aforementioned linker, because of sheet contraction along the width contract, basically does not contribute to the load carrying process. In contrast and as illustrated in
Figure 5c, during the loading along the armchair direction, all diacetylene linkers in these systems stretch and contribute to the load transfer, resulting in a higher tensile strength and stretchability as well.
We next explored the possible efficiency of the BTX-GDY monolayers for overall photocatalytic (PC) water splitting reactions. In the PC water splitting process, light is utilized by a semiconductor with specific electro-optical properties to split water into H
2 and O
2 under mild conditions. An appropriate semiconductor for spontaneous PC water splitting, in addition to a large band gap (>1.23 eV), is also supposed to yield band edge positions straddling the water redox potentials (H
+/H2 (HER): −4.03 eV vs. vacuum, O
2/H
2O (OER): −5.26 V vs. vacuum, both at pH 7). As shown in
Figure 2, only for the BTF- and BTT-GDY monolayers under a neutral condition, the VBM level is less than the oxidation potential of the OER, and the CBM is located higher than the reduction potential of the HER, suggesting they might be promising photocatalysts for overall PC water splitting. A semiconductor with an appropriate band offset for PC water splitting is also expected to show good carrier mobilities able to enhance the efficiency of the energy conversion process. To estimate the electron and hole mobilities within the two promising candidates of BTF- and BTT-GDY, we employed the conventional deformation potential theory.
Table 2 lists the calculated effective masses (
), deformation potential constants (
C2D) and estimated electron and hole mobilities (
µ) for these two polymers. Because of the hexagonal symmetries of the BTF- and BTT-GDY, we only dilated their primitive cells along the zigzag direction in the range of ±1.0% and calculated the band edge positions at different degrees of dilation. The predicted acoustic phonon-limited electron mobilities are 6.70 and 4.79 cm
2 V
−1 s
−1 for the BTF- and BTT-GDY monolayers, respectively, which are appreciably larger than corresponding values for holes, with almost flat VBs, 0.42 and 0.02 cm
2 V
−1 s
−1, respectively. The estimated carrier mobilities for holes, however, are, as expected, significantly larger than the electron mobilities and are 855 and 451 cm
2 V
−1 s
−1 for the BTF- and BTT-GDY monolayers, respectively, indicating that holes are dominant carriers in these nanomaterials. These values are even larger than those predicted for the MoS
2 monolayer using the same theory and approximations (~200 cm
2 V
−1 s
−1 [
35]). It is also worth noting that the experimental calculations for the BTT-GDY also reveal much higher hole mobility than electron mobility (2.34 and 3.35
× 10
−2 cm
2 V
−1 s
−1 for holes and electrons, respectively [
19]). These findings are promising, because a large difference between electron and hole mobilities is appealing for efficient electron–hole separation and higher PC process efficiency.
Other than a large band gap, an appropriate band offset and good carrier mobilities, a semiconductor photocatalyst needs to present a good light absorption within the visible region. We calculated the absorption coefficient (α) for the BTP-, BTT- and BTF-GDY monolayers along the in-plane polarization as a function of wavelength with the HSE06-based RPA method, and the obtained results are plotted in
Figure 6. Because of the hexagonal symmetries of BTX-GDY, only the optical properties along the zigzag direction are reported. The absorption coefficients of two different allotropes of the C
3N
4 monolayer, s-triazine-C
3N
4 (
gt-C
3N
4) and tri-s-triazine-C
3N
4 (
gh-C
3N
4), which are famous for their photocatalytic activity, are also reported. The first peak of α for the BTT-GDY nanosheet occurs at a wavelength of 470 nm while the corresponding first peak for BTP-GDY and BTF-GDY monolayers shifts to the shorter wavelengths (blue shift), occurring at wavelength of 450 nm. These results show the first absorption peak for these novel 2D materials appears in the visible range of light (blue-light wavelength). The first absorption peak for the
gh-C
3N
4 and
gt-C
3N
4 monolayers occurs at wavelengths of 401 and 372 nm, respectively, which are in the violet-light wavelength. It can also be seen that the absorption coefficients for the BTP-, BTT- and BTF-GDY monolayers in the wavelength range between 375 and 600 nm are larger than those of C
3N
4 monolayers. The high absorption coefficient (~10
5 cm
−1) for these novel 2D systems within the visible range of light is highly desirable for visible-light-driven optoelectronic applications.
Last but not least, graphdiyne nanomembranes are well-known to show outstanding efficiency as anodic materials in metal-ion batteries [
2,
3,
4,
36,
37]. In this regard, owing to their light atomic weight and nanoporous structures, graphdiyne nanosheets may show ultrahigh storage capacities for various metal ions. We thus next briefly examine the Li-ion storage capacity of the BTHP-GDY, BTT-GDY and BTF-GDY monolayers. To this end, first, the strongest binding site for a Li adatom adsorption was identified, according to the adsorption energy,
Ead:
Here,
EM is the total energy of the pristine monolayers,
ELiM is the total energy of the monolayers after the adsorption of a Li atom, and
ELi is the per atom energy of the bulk Li. Eight different original adsorption sites were considered, and after the energy optimization with the conjugate gradient method, their corresponding adsorption energies were calculated. The adsorption energy of the Li adatoms over the strongest binding sites for the BTT-GDY and BTF-GDY monolayers was found to be +0.63 and +0.67 eV, respectively. The predicted positive adsorption energies reveal that the Li adatoms prefer to form clusters rather than adsorption over the BTT-GDY and BTF-GDY nanosheets’ surfaces. Nitrogen doping is a well-known approach to efficiently boast the anodic performance of carbon-based materials [
38,
39,
40]. Interestingly, as shown in
Figure 7a, the BTHP-GDY nanosheet with active N atoms was found to yield strong adsorption with Li atoms. In order to estimate the storage capacity, the content of Li adatoms over the surface must be increased. To simulate such a process, first all the strongest adsorption sites are gradually filled, which requires six Li atoms over the BTHP-GDY unitcell. As it is clear from
Figure 7a, the first and second strongest adsorption sites are very close, and because of the repulsive forces between adatoms, the second position is neglected in this step. On this basis, next, all third strongest adsorption sites over the BTHP-GDY unitcell are filled, which includes 18 Li adatoms. In the final step, using the energy-minimized structure with 18 Li atoms, an additional 12 adatoms are included over the diacetylene linkers on both sides, and 4 Li atoms are placed on both sides of central full-carbon hexagonal rings and consequently the full geometry optimization is carried out. For every Li content, the average adsorption energy is calculated [
4], which is illustrated in
Figure 7b. As an interesting finding, the average adsorption energy of the BTHP-GDY monolayer with 34 Li adatoms in the unitcell was predicted to be −0.1 eV, which reveals that the metal atoms can still efficiently adsorb over the surface. Bader charge analysis calculations show that upon the adsorption of 34 Li adatoms, they transfer their valence electron to the substrate and become ionized. This finding is consistent with the experimental process, in which the Li ions diffuse from the cathode through the electrolyte and finally adsorb over the anode. It should also be noted that the simulated adsorption process of Li ions is a simplified and rather rough approach. By taking into account that there exists a high probability that other configurations with lower energies but higher adsorbed Li-ion content can form, it is thus clear that herein, estimated storage capacity can only be considered as a lower bound. On this basis, we theoretically predict that the BTHP-GDY nanosheets might be able to yield a Li-ion storage capacity over 2400 mAh/g, which is more than six-fold higher than that of the commercial graphite, at 372 mAh/g [
4]. This finding once again reveals the outstanding capacity of the N-based GDY nanosheets for application in rechargeable batteries.