Introduction

Given the ever-increasing population growth, reduced fossil fuels, and, more importantly, greenhouse gas emissions, the development of clean energy sources is indispensable [1]. Many developed countries are looking for alternative sources of energy. Among the studied cases, hydrogen has received much attention because of its high abundance, performance, quality, and lacking pollution [2,3,4]. Despite these advantages, the impossibility of hydrogen storage in environmental conditions has prevented the widespread use of this fuel [5].

In this regard, there are three methods of compression, liquefaction, and storage on a solid bed for the storage of hydrogen [6, 7]. Research has shown that suitable hydrogen adsorption energy for H2 ranges from − 0.2 to 0.7 eV/H2 for adsorbed and desorbed states in environmental conditions, hence storing hydrogen in a solid bed seems to be the most appropriate method [8, 9].

Various solid matters, such as graphene [5, 10, 11] and silicone [12,13,14,15], have been investigated as a solid bed. Graphene is a two-dimensional substance consisting of carbon atoms with a honeycomb form. Because of its excellent properties, high thermal conductivity, and excitability in electrical conductivity, this material has attracted much attention compared with other carbon materials. However, the absorption energy of hydrogen molecule on pure graphene surface is weak [9, 16]. Addition of intermediate metals is an effective way to improve absorption energy [8,9,10,11,12,13]. Metals that have been studied include alkali metals (AMs) (Li, Na, and K), alkali earth metals (AEMs) (Be, Mg, and Ca), transition metals (TM) (Sc, Ti, and V), and a simple metal (Al).

In metal elements, binding between transition metals and hydrogen molecule results from Kubas interaction, which emanates from the hybridization between H2 molecule σ∗ antibonding orbital and d orbitals of transition metals [17, 18]. The binding energy between H2 molecule and TM is about 0.4 eV, which is suitable for the storage of hydrogen [15]. These metals, however, tend to accumulate on the graphene surface, which reduces the storage capacity of hydrogen [19]. Studies have shown that the formation of pores not only prevents the accumulation of intermediate metals on the graphene surface but also creates greater binding sites for the hydrogen molecule leading to its increased storage capacity [19,20,21].

Due to the benefits of intermediate metal atoms and the role of pores in graphene, it has been used to improve hydrogen storage. Several studies have been conducted in this regard. Yuan et al. [21] reviewed Y decoration on PG through the evaluation of adsorption energy using the CASTEP code. The hydrogen molecule could be absorbed around an atom of Y with an average energy of − 0.297 eV. Chen et al. [9] studied hydrogen storage on SC-PG based on DFT calculations. For a Sc atom, the most stable place is the center of the hexagonal carbon ring. Hydrogen can be absorbed around a Sc atom, with a mean binding energy of − 0.429 eV. The ability to absorb hydrogen molecules on Ti-PG has been studied recently. Ti atom was placed over the center of hexagonal carbon ring, so that Ti could highly be adsorbed on PG. Four hydrogen molecules around Ti atom were absorbed with a mean absorption energy of − 0.457 eV [16].

In the present study, the problem was improved by choosing the topology defect that holds the symmetry in a single layer of graphene and evaluates the hydrogen molecule adsorption. The structure was made by changing the size of this topological defect to keep the symmetry. The most stable adsorption site of Sc atoms is determined on four different sizes of PG by considering the mode of action of Sc-PG interplay. Subsequently, a discussion is presented on the adsorption ability of H2 molecule on Sc-decorated PG system, followed by comparing the storage capacity and the adsorption mechanism of these four PGs. Eventually, the adsorption energy of H2 molecules on PGs in the absence of Sc is examined for further understanding of Sc atom impacts.

Calculation details

All calculations were done utilizing the FHI-aims code. FHI-aims was programmed and set up to allow efficient all-electron calculations for any system type. The generalized gradient approximation (GGA) and PBE function were used to calculate the energy. The polarized spin calculations) collinear) were done in this work, along with using the light settings for all calculations. The entire atoms will be able to relax in all calculations, along with full optimization of the whole structure. The computational unit cell comprises a 5 × 5  of graphene cell with a vacuum of 15 Å. The periodic boundary conditions are applied to calculate the PG unit cell. Based on Tkatchenko and Scheffler, van der Waals (vdW) correction was used to investigate the role of vdW interaction in all computations by the use of a 5 × 5 × 1 k-point mesh. The Pulay algorithm was used to improve the relative ZORA approximation and to achieve a fast and stable convergence in the self-consistent cycle. A Gaussian occupation broadening was selected with a width of 0.05 eV.

Convergence criteria of 10−5eV for the electron density, 10−3eV for the eigenvalues, and 10−6 eV were applied for total energy of the system. The adsorption energies of Sc atom on PG structures are obtained by Eq. (1):

$$ {\overline{E}}_b=\left[\ {E}_{\mathrm{Sc}+\mathrm{PG}}-{E}_{\mathrm{PG}}-\mathrm{n}{E}_{\mathrm{Sc}}\ \right]/n $$
(1)

where ESc+PG, EPG, and ESc are total energies of the system with Sc atoms, PG layer, and a free Sc atom, respectively, and n is the number of adsorbed Sc atoms. The adsorption energy and mean adsorption energy of each H2 molecule on Sc-decorated PG system are calculated by

$$ {E}_{ab}=\left[{E}_{i{\mathrm{H}}_2+\mathrm{Sc}+\mathrm{PG}}-{E}_{\left(i-1\right){\mathrm{H}}_2+\mathrm{Sc}+\mathrm{PG}}-{E}_{H_2}\right] $$
(2)
$$ {\overline{E}}_{ab}=\left[{E}_{i{\mathrm{H}}_2+\mathrm{Sc}+\mathrm{PG}}-{E}_{sc+\mathrm{PG}}-i{E}_{{\mathrm{H}}_2}\right]/i $$
(3)

where\( {E}_{\mathrm{i}{\mathrm{H}}_2+ Sc+ PG} \), \( {E}_{\left(i-1\right){\mathrm{H}}_2+ Sc+ PG} \), and \( {E}_{{\mathrm{H}}_2} \) are total energies of the system with i H2 molecules adsorbed, the (i−1) H2 molecules adsorbed, and free H2 molecule, respectively. PG has two types of carbon atoms, C1 and C2, the former is connected to two carbons and one hydrogen atom, and the latter is linked to three carbon atoms. The bond distances of C1–H, C2–C2 and C1–C2 are 1.085, 1.427, and 1.395 Å, respectively. Our results are in agreement with those of Li [22], who reported observations with GGA-PW91 function using DMol platform, indicating that our computational results are reliable.

Results and discussion

Electronic structure of Sc-decorated PG

Initially, the adsorption of Sc atom was considered on four different sizes of PG. According to Fig. 1, h1, h2, b1, b2, t1, and t2 are six sites for adsorption of a Sc atom on PGs, namely the center of C ring, the center of a half C ring, the bridge of C1–C2 bond, the bridge of C2–C2 bond, the top of C1 atom, and the top of C2 atom, respectively. Examination of the adsorption energies for different positions of the Sc atom on PGs reveals that the center of a hexagonal ring (h1) is the preferred adsorption site for a Sc atom on PGs.

Fig. 1
figure 1

Six primary absorption sites for a Sc atom on PG1. Gray and white balls indicate C and H atoms

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The configuration in the presence of Sc atom after relaxation is displayed in Fig. 2. Accordingly, the position of Sc atom is slightly deformed with respect to the center of the C ring, with adsorption energies of − 2.654, − 2.693, − 2.909, and − 2.696 eV for a Sc atom on PG1, PG2, PG3, and, PG4, respectively.

Fig. 2
figure 2

The optimized geometry structure of a Sc atom–decorated PGs. (a) Sc-PG1, (b) Sc-PG2, (c) Sc-PG3, and (d) Sc-PG4. Gray, white, and red balls are C, H, and Sc atoms, respectively

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The adsorption energy of Sc atom on pure graphene was also calculated to explain the effect of pores in graphene layer on the adsorption of Sc atom. The adsorption energy of Sc atom on 5 × 5 supercell of graphene is − 1.892 eV, which corresponds to that of Chen [23]. The adsorption energy of Sc atom on PGs is clearly higher than that of Sc atom on pure graphene. Table 1 lists the analysis of Mulliken atomic populations for Sc atom on four PGs. It shows that the partial charge of the Sc atom are donated from the 4s orbital to the PGs, which increases the negative charge of carbon and forms an electric field between Sc and PG layers. This results in back donations part of the charge from PG to the 3d orbital of Sc atom via hybridizations between C and Sc atoms. Increasing the pore size is associated with rising the amount of charge transfers from Sc atom.

Table 1 Mulliken population analysis of Sc atom on PG1, PG2, PG3, and PG4 structures
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The partial densities of states (PDOS) of Sc atom and C atom of Sc-PG system were examined to realize well the interplay between Sc atom and PGs. For all considered systems (Fig. 3), there is overlap between C 2p and Sc 3d orbitals around Fermi level, which indicates a robust hybridization between C and Sc atoms. Consequently, the polarization and the hybridization can contribute to Sc adsorption. Figure S1 depicts the band structures for all PGs in Supporting Information. PG1 has almost the same electronic structure as graphene, and Dirac cones are at the k-points in both structures. PG2 disturbs Dirac points, and Dirac cone forms at the M-point; therefore, the electronic structure of PG2 is different from that of graphene. PG3 and PG4 break the lattice symmetry of pure graphene, and a band gap is formed at the Dirac point. For PG3, the Fermi level is the middle of the band gap, which shifts toward the valence band for PG4. It is also seen that the adsorption of Sc atom results in an n-type doping of the PGs in which the Fermi level shifts toward the conduction band.

Fig. 3
figure 3

PDOS of Sc (s, p, and d orbitals) and C (s and p orbitals) atoms in the system of Sc atom adsorbed on (a) PG1, (b) PG2, (c) PG3, and (d) PG4. The Fermi level was fixed to 0

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Subsequently, the states of second Sc atoms adsorbed on PG structures are considered to study the accumulation of Sc atoms. Analysis of the energy indicates that the most preferable adsorption position is the center of C hexagon on the perimeter of PGs. As revealed by the present computations, the binding energy of the second Sc atom for all PG structures is stronger than the first one, resulting from the electric field stimulated by the first Sc atom. However, the cohesive energy of the Sc atom is greater than its binding energy on PGs, and the two Sc atoms lie closely to one another. This may be a trend of clustering among Sc atoms; therefore, only one Sc atom on PGs on the single side should be studied to avoid the clustering of Sc atoms.

Adsorption of H2 molecules on Sc-decorated PGs

The capacity of hydrogen storage for Sc decorated on PG1, PG2, PG3, and PG4 is discussed here. The first H2 molecule favors adsorption on the bridge of C–C bond as shown in Fig. 4. In Table 2, the adsorption energies of the first H2 molecule of the four structures are − 0.568, − 0.553, − 0.509, and − 0.566 eV, respectively. Besides, the bond distance of H–H is extended from 0.753 Å of the isolated molecule to the range of 0.830–0.838 Å [18], indicating a robust adsorption of H2 on Sc-PGs system.

Fig. 4
figure 4

Optimized geometries of a Sc atom–decorated PGs with one H2 adsorbed there on. (a) PG1, (b) PG2, (c) PG3, and (d) PG4, respectively

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Table 2 Estimated binding energies and bond distances of one hydrogen molecule adsorbed on Sc-decorated PG1, PG2, PG3, and PG4, respectively
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Analyses of charge density difference, partial density of states, adsorption energy of PGs, and Sc-decorated PG system with one H2 adsorbed are performed to examine the impact of Sc atom on the adsorption of H2 molecule. The charge transfers of the PG and Sc-PG systems are observable via charge density difference (∆ρ), so ∆ρ is presented as

$$ \Delta \rho ={\rho}_{{i\mathrm{H}}_2+\mathrm{PG}}-{\rho}_{i{\mathrm{H}}_2}-{\rho}_{\mathrm{PG}} $$
(4)
$$ \Delta \rho ={\rho}_{{i\mathrm{H}}_2+\mathrm{Sc}+\mathrm{PG}}-{\rho}_{i{\mathrm{H}}_2}-{\rho}_{\mathrm{Sc}+\mathrm{PG}} $$
(5)

where \( {\rho}_{{i\mathrm{H}}_2+\mathrm{Sc}+\mathrm{PG}} \), \( {\rho}_{{i\mathrm{H}}_2+\mathrm{PG}} \),\( {\rho}_{i{\mathrm{H}}_2} \), ρSc + PG, and ρPGdenote the charge density of the total Sc-PG system, the charge density of total PG system, the charge density of i adsorbed H2 molecules, the charge density of Sc-decorated PG system, and the charge density of PG system, respectively. The charge density difference of the first H2 molecule adsorbed on PG structures is displayed in Fig. 5, where the blue and red colors denote electron accumulation and depletion sites, respectively. For Sc-PG systems, Fig. 5(a) depicts a clear charge accumulation between H2 molecules and Sc atom. In addition, charge depletion and accumulation observed at either side of the H2 molecule indicate the polarization of H2 molecule, thereby raising the H2-PG interplay because of the Kubas interaction [18, 24,25,26,27]. Values for the isosurface of the four structures are 0.006, 0.007, 0.007, and 0.007 e/Å3, respectively. The charge density difference of the first H2 molecule adsorbed on PGs without the presence of Sc atom (Fig. 5(b)) indicates that the charge distribution is nearly on the pore center and the isosurface has a value of about 0.000 e/Å3 for all the structures. Furthermore, from Fig. 5(a), it is seen that the existence of Sc atom causes the accumulation of charge on the perimeter of pores.

Fig. 5
figure 5

The isosurfaces of charge density differences for one H2 adsorption on PGs. The blue and red colors characterize electron accumulation and depletion sites. (a) Sc-decorated PGs with one adsorbed H2 and (b) PGs with one adsorbed H2

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From PDOS of the first H2 molecule adsorbed on PGs and Sc-decorated PG system (Fig. 6) and the Supporting Information (Figs. S2, S3, and S4) for all the PGs, it can be observed that H2 molecule in PG system has a weak interaction because there is no overlap between H2 and PGs. In Sc-decorated PG systems, on the other hand, several overlaps exist in the main peaks of H21s, Sc3d, and C2p orbitals around the Fermi level, suggesting strong interactions between H2, Sc, and PGs. Among all possible structures, PG3 has a weak interaction with H2 molecule because of small sharp peaks around the Fermi level. Figure 7 displays the adsorption energy of the first H2 on Sc-PGs and that of the first H2 on PGs without the presence of Sc atom. The binding energies for Sc-PG system ranges from 0.2 to 0.7 eV/H2 fulfilling the necessity that H2 molecule is capable of adsorption and desorption under surrounded settings [28]. In addition, the adsorption energy of the first H2 on PG3 is less than that of PG1, PG2, and PG4, as displayed in Fig. 8(a). The adsorption energy of PGs without the presence of Sc atom is not in the desirable range (Fig. 8(b)). Therefore, the existence of Sc atom would reinforce the adsorption energy of H2 molecule on PGs.

Fig. 6
figure 6

PDOS for H2 1s (green line), Sc 3d (red line), and C2p (blue line) orbitals when one H2 molecule is adsorbed on (a) Sc-PG1 and (b) PG1 without the presence of Sc atom. The Fermi level is located at 0

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Fig. 7
figure 7

Calculated binding of the first H2 molecules on (a) Sc-decorated PGs and (b) on PGs without the presence of Sc atom

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Fig. 8
figure 8

Optimized atomic structure of Sc- PG1 system with (a) one, (b) two, (c) three, (d) four, and (e) five hydrogen molecules attached

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Figure 8 presents the post-relaxation configuration for PG1 in the event of extra H2 molecules adsorption on Sc-PG structure. For the adsorption of H2 on Sc-decorated PG2, PG3, and PG4, the optimized geometries resemble that on Sc-decorated PG1. H2 molecules prefer symmetrical absorption around Sc atom. As shown in Fig. 8, the first four H2 molecules are bonded to Sc atom, and the fifth H2 molecule moves to an upper layer without forming a bond with Sc atom. Furthermore, the adsorption energies of the fifth H2 molecule on PGs are − 0.167, − 0.114, − 0.159, and − 0.169 eV, respectively, being incapable for hydrogen storage uses [29]. Hence, an uppermost number of four H2 molecules are absorbable on Sc-PG system, and average adsorption energies for PG1, PG2, PG3, and PG4 are − 0.547, − 0.539, − 0.546, and − 0.553 eV/H2, respectively, being higher than a mean adsorption energy of − 0.429 eV/H2 reported by Chen [9] (Table 3).

Table 3 Absorption energies (ΔE), average adsorption energies (ΔĒ), bond lengths of hydrogen, and charges of Sc atom (ΔqSc) on Sc-PG1 system
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Figure 9 displays the PDOS of H2 1s and Sc 3d orbitals upon adsorption of 1–5 H2 molecules on Sc-decorated PG1. In addition, analysis of the PDOS of H2 molecules on Sc-decorated PG2, PG3, and PG4 demonstrates a similar adsorption behavior in all the four structures. It is obvious that the peaks of H2 1s and Sc 3d orbitals overlap within − 10 to − 8 eV, suggesting a strong hybridization between orbitals of H2 1s and Sc 3d. Furthermore, the adsorption of 1~4 H2 molecules also reveals that there is a hybridization between H2 and Sc atom around − 1 eV, which can be a reason for the lack of a strong interplay of the fifth H2 molecule with Sc atom.

Fig. 9
figure 9

PDOS of H2 1s (red line) and Sc 3d (blue line) orbitals for adsorption of 1–5 H2 molecules in Sc-PG1. The Fermi level is fixed at zero

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Figure 10 summarizes the adsorption energy of each H2 molecule on PG1, PG2, PG3, and PG4 with rising number of H2 molecules. It is observed that the adsorption energy of the second H2 is greater than that of the first H2, and the adsorption energy of the third H2 is weaker than that of the forth H2 on all the four structures. As this result is understood from PDOS (Fig. 9(a) and (b)), the band expansion of the H2 molecular level below the Fermi energy (around − 9 eV) shows the interplay between H2-H2 molecules, which in turn could elevate the binding energies of the second H2, being in full agreement with previous reports [30,31,32]. Additionally, analysis of the H–H bond length (Supporting Information in Table S1) can justify the relatively weak adsorption of the third H2. Figure 9 also displays that the adsorption energy of H2 molecules on Sc-decorated PGs (except for PG1) rises regularly with growing the pore size, which may be due to the symmetry of PG structures. However, the binding energies of H2 molecules on Sc-adorned PG4 are uppermost among the four studied structures. Our calculations show that the system with a higher overlap around the Fermi level is suitable for hydrogen storage.

Fig. 10
figure 10

A comparison of adsorption energies for five hydrogen molecules on four different sizes of porous graphene

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Conclusions

The adsorption of Sc atom on four different sizes of porous graphene and of H2 molecules on Sc-decorated PG system were studied using FHI-aims code. A very slight deformation of PGs suggests that Sc atom prefers to be adsorbed on the central area over the C hexagon, and that Sc atom has a strong adsorption on PGs. Charge density difference, partial density of states, and adsorption energy were analyzed to examine the effect of Sc atom on the adsorption of H2 molecule, showing that the presence of Sc atom would strengthen the adsorption energy of H2 molecule on PGs. Four H2 molecules are adsorbable around Sc atom for the single side of PGs. According to the analysis of the partial density of states, the adsorption of H2 molecules on the PG systems is related to orbital hybridization among Sc, H, and C atoms, as well as to the polarization between Sc and H atoms. The mean adsorption energies of H2 molecules on Sc-decorated PG4 were uppermost among the four studied structures. Accordingly, the adsorption energy of H2 molecules on Sc-adorned PG system rises steadily with increasing the pore size.