Introduction

Graphene is a two-dimensional layer of carbon atoms in honeycomb lattice structure, arranged with \(sp^2\)-hybridization (Geim and Novoselov 2007). With a number of wonderful properties, graphene has a long history (more than 60 years) of being studied first theoretically in 1947 (Wallace 1947) and later on experimentally in 2004 (Novoselov et al. 2004). Besides the attractive physical properties as a thin, strong and stretchable material (Geim 2009), graphene has interesting electronic/magnetic properties which enhances its potential use in academic and industrial world.

Graphene, its derivatives and other similar two-dimensional compounds have become frontier area of research activities, and also the materials of commercial interest in recent days (Medeiros et al. 2010; Johll et al. 2009; Thapa et al. 2011; Chan et al. 2008; Karki and Adhikari 2014; Ci et al. 2010; Kaloni et al. 2011; Park and Louie 2010; Kaloni et al. 2013; Esquinazi et al. 2003; Mukherjee and Kaloni 2012; Ugeda et al. 2010; Singh et al. 2013; Kaloni et al. 2014; Saha et al. 2009; Kaloni et al. 2012; Wu and Yang 2012; Dai et al. 2009; Choi and Jhi 2009). Graphene, hexagonal boron nitride (h-BN) and their doped structures have been studied by a number of research groups to see their potential applications in advanced electronics and optics (Ci et al. 2010; Mukherjee and Kaloni 2012; Singh et al. 2013). Also the multilayer graphene and other graphitic arrangements with electron donor/acceptor atoms, at varying concentrations, are covered to see their modified magnetism, and band structures (Kaloni et al. 2011, 2014, 2008; Park and Louie 2010; Kaloni et al. 2012; Ataca et al. 2008). The varying band gaps of these new materials and other carbon nanostructures include extensively a wide range of conductivity with all the potential applications in semiconductor (Mukherjee and Kaloni 2012; Kaloni et al. 2014), conductor (Ataca et al. 2008) to superconductor (Ugeda et al. 2010). The easily tunable structural stability, band structure and other electronic/magnetic properties, and also photonic and vibrational properties of graphene have been achieved through a number of techniques like: via vacancies (Kaloni et al. 2013; Ugeda et al. 2010; Singh et al. 2013; Kaloni et al. 2012), atomic/molecular doping (Medeiros et al. 2010; Johll et al. 2009; Thapa et al. 2011; Chan et al. 2008; Karki and Adhikari 2014; Saha et al. 2009; Wu and Yang 2012), functionalization (Wood et al. 2012; Ulman et al. 2014), irradiation (Esquinazi et al. 2003) and external fields (Park and Louie 2010). The modified compounds carry potential applications over many dimensions like in electronics (Geim 2009; Novoselov et al. 2005, 2007), spintronics (Palacios et al. 2008; Ding et al. 2011), chemical sensors (Saha et al. 2009), and energy storage (Pumera 2011). Graphene with the adsorption of relevant metal atoms, clusters and other functional groups as impurities have been seriously considered and studied as substrates for gaseous adsorption (Wu and Yang 2012), including energy carrying gases methane and hydrogen (Wood et al. 2012; Ulman et al. 2014).

Hydrogen, either in gaseous form or as a component of abundant compounds like water and hydrocarbons, is a widely available energy resource in the Earth and extra-terrestrial planets. As hydrogen releases only the water vapour when it burns, it is considered as one of the clean and green sources of energy. In spite of having many advantages to be a potential gas for future energy carrier, user friendly storage and transportation have become one of the major challenges. There are a number of traditional techniques which have been practiced to store natural gases and hydrogen, like in tanks under high pressure (Compressed natural gases, CNG) (Burchell and Rogers 2000; Duren et al. 2004) and in liquid form at low temperature (Liquefied natural Gases, LNGs) (Nakanishi and Reid 1971) or in chemical hydrides (Orimo et al. 2007). The techniques, however, are not user friendly due to weight, space and economy concerned perspectives and also they display the risk of leakage and explosion (Chen et al. 1999). Adsorption of hydrogen in different porous materials (Ma et al. 2008) and carbon nanomaterials including activated carbons is some of the highly searched techniques in recent time (Pan et al. 2008). It has been reported that an appreciable amount of hydrogen could be stored at low temperature and the continuous development is going on to develop the devices which are suitable to store hydrogen at operating conditions. Among the carbon nanotubes and graphene allotropes as substrates for hydrogen adsorption, adatoms including alkali elements have performed catalytic effect in dissociating the H\(_2\) molecule, promoting atomic adsorption and enhancing binding strength at moderate temperature and ambient pressure (Kwon 2010; Chen et al. 2008; Chandrakumar and Ghosh 2008). The United State’s department of energy (US, DOE) has set targets of about 6 wt% (Durgun et al. 2008) and adsorption energy range (0.2–0.7 eV) (Zhou and Willians 2011) for hydrogen storage in practical applications like vehicular transportation. Approaching towards these goals by designing the proper substrate will explore interesting scientific and commercial implications.

The adsorption of alkali metal atoms on graphitic materials has been widely studied to see the interaction between them (Medeiros et al. 2010; Liu et al. 2011). Being as electron donor atoms, electronic charge transfers from adsorbed alkali atoms (Li, Na and K) to graphene, which causes change in band structure and neutrality of the new materials (Mukherjee and Kaloni 2012). The properties of atom-adsorbed graphene can also be tuned using mechanical strain (Zhou et al. 2010). Among the alkali metal atoms, Li with its light weight has been previously tested to metalize graphene, which enhances hydrogen storage capacity up to 12.8 wt% (Ataca et al. 2008). Having similar electronic properties and slightly higher molecular weight, Na could be a good option for catalyzing graphene to store H\(_2\)-moleculars. However, in our best knowledge, this element neither as an isolated atom nor as a cluster in carbon nanostructures has been checked for enhancing gaseous adsorption. In the present work, we first see the nature of interaction between the Na atom with monolayer graphene and then Na-graphene system is used to study its binding strength for H\(_2\)-moleculars.

Computational details

Density functional theory (DFT)-based first-principles calculations (Hohenberg and Kohn 1964; Kohn and Sahm 1965) are carried out to investigate the structural stability and electronic/magnetic properties of graphene and single sodium atom added graphene. The optimized geometry of 3 \(\times\) 3 Na-added graphene, based on compromise between the size effect and the computational cost, is then used as substrate to study the adsorption behavior of a number of H\(_2\)-molecular. The calculations are performed by incorporating van der Waals (vdW) interactions (Klimes and Michaelides 2012) via London dispersion effects in DFT-D2 approach, implemented with the quantum ESPRESSO package (Giannozzi et al. 2009). The algorithm has used Rappe–Rabe–Kaxiras–Joannopoulos (RRKJ) model of ultrasoft pseudopotential to account the interaction between the ion cores and valence electrons and generalized gradient approximation (GGA) formalism to treat the electronic exchange and correlation effects, as described by Perdew–Burke–Ernzerhof (PBE) (Perdew et al. 1996), for all the species (C, Na, H). The semi-core states: \(2p3s\) of sodium, \(1s\) of hydrogen and \(2s2p\) of carbon atoms are treated explicitly as valence in the pseudopotential description of the corresponding atoms. The inner cores for carbon and sodium, on the other hands, are included as ion cores by the corresponding pseudo potentials Quantum (2014).

A hexagonal unit cell with the basis of two atoms in honeycomb lattice structure was initially constructed using experimental value (Castro Neto et al. 2006, 2009). The structure was then optimized with respect to lattice parameter (a), cut off energy of the plane waves and the number of k-points along x and y axes, respectively. Based on these convergence tests, a plane wave basis set with the kinetic energy cutoff 35 Ry is used for the expansion of the ground state electronic wave function. The plane waves are chosen to have a periodicity compatible with the periodic boundary conditions of the simulating cell. In case of lattice constant, we have used the calculated value (2.46 Å) which comes to agree with the experimental value for the planner sides of the unit cell, whereas the dimension along z-direction is kept large enough (20 Å) to avoid the interaction between the graphene layers and also to provide the space for upcoming hydrogen adsorption calculations. Similarly, we have used 15 \(\times\) 15 \(\times\) 1 mesh of k-points from convergence test for the unit cell calculations (Fig. 1).

Fig. 1
figure 1

The convergence of total energy with respect to the lattice parameter (a), cut off energy (b) and k-points (c)

Full size image

The Na-adsorbed graphene system is modeled using single Na atom in the 2 \(\times\) 2, 3 \(\times\) 3 and 4 \(\times\) 4 supercell of graphene containing 8, 18, and 32 number of carbon atoms. In this work, the adsorption of sodium on graphene is performed at three different occupation sites: the Top (T) site directly above the carbon atom, the Hollow (H) site at the center of hexagon and the bridge (B) site at the midpoint of the C–C bond (Fig. 2). For each adsorption site of the adatom-graphene system, the foreign atom (Na) and C atoms on graphene are relaxed in all x, y and z directions. To estimate the binding energy of Na, the calculations for the isolated Na, isolated graphene and Na-adsorbed graphene system are performed in same-sized graphene supercell.

Fig. 2
figure 2

Three occupation sites top (T), bridge (B) and hollow (H) of the graphene sheet are shown

Full size image

During the process of calculations, we have first performed the relaxation of initial geometries to obtain the optimized structures. The systems thus calculated are allowed fully to relax using BFGS (Broyden–Fletcher–Goldfarb–Shanno) scheme until the total energy changes between two consecutive self-consistent field steps are less than \(10^{-4}\) Ry and force acting is less than \(10^{-3}\) Ry/Bohr. The first Brillouin zone of graphene is sampled in the reciprocal space using the Monkhorst–Pack scheme with an appropriate number of k-points, as described by the convergence test. Smearing was incorporated to aid the convergence. We have used ‘Marzarri–vanderbilt’ method (Marzari et al. 1999) or cold smearing with a small Gaussian spread of 0.001 Ry. Furthermore, we have chosen ‘david’ diagonalization method with the mixing factor 0.6 for self consistency. Spin polarized calculations are accommodated to study the magnetic properties of the systems. For the density of states (DOS) calculations of pure graphene and sodium-adsorbed graphene systems, we have used 3 \(\times\) 3 supercell of graphene with a denser mesh of (15 \(\times\) 15 \(\times\) 1) k-points.

We have also investigated the stability of hydrogen adsorption on sodium added graphene system. As a first step, H\(_2\)-molecular was relaxed on similar box of 3 \(\times\) 3 graphene supercell and then defined its optimized geometry accordingly. The bond length of H\(_2\) molecule calculated in this way has been found as 0.75 Å which agrees with the previously reported value (Arellano et al. 2012). Due to the periodicity of the crystal, the distance between two H\(_2\)-moleculars of nearest supercells becomes equal to the cell dimension (i.e., 9.299 Bohr) which is large enough compared to the molecular bond length of H\(_2\) (i.e., 1.418 Bohr). This situation ensures that there is no interaction between two H\(_2\)-moleculars located at adjacent supercells.

Results and discussion

In the present work, we study the adsorption of single sodium atom on three different occupation sites of 2 \(\times\) 2, 3 \(\times\) 3 and 4 \(\times\) 4 supercells of pure graphene. Furthermore, we extend our work to study the adsorption of H\(_2\)-moleculars on pure graphene and Na-adsorbed graphene.

Adsorption of sodium atom on graphene

We discuss structural, electronic and magnetic properties of pure and sodium added graphene in this section.

The binding energy of sodium atom on graphene sheet is calculated using relation,

$$\Delta E = E_{{{\rm {Na}}}} + E_{{\rm {G}}} - E_{{{\rm {G + Na}}}}$$
(1)

where \(E_{\rm Na}\), \(E_{\rm G}\) and \(E_{\rm G+Na}\) are the ground state energies of sodium atom, pure graphene and sodium-adsorbed graphene systems in the fixed supercells, respectively. The positive binding energy from Eq. (1) indicates stable configurations. We have defined the most preferred sites/geometries of the systems for 2 \(\times\) 2, 3 \(\times\) 3, and 4 \(\times\) 4 supercells separately by taking information of the largest binding energy values from these calculations.

The adatom height (h) is defined as the difference in \(z\) coordinate of the Na and an average of \(z\) coordinates of the C atoms in graphene layer. We have also calculated the distance (\(d_{\rm Ac}\) ) between the adsorbed atom and its nearest carbon atom. The adsorption of sodium on graphene produces a small distortion which is quantified by computing the maximum deviation in \(z\) direction of C atoms in graphene layer from their average initial position. This distortion of the graphene layer upon the adsorption of a foreign atom (Na in this case) is also calculated in terms of change in dihedral angles.

The binding energy of sodium atom on hexagonal 2 \(\times\) 2 graphene supercell, containing 8 carbon atoms, has been calculated as \(-\)0.29, \(-\)0.38, \(-\)0.39 eV on H, B, and T sites, respectively. This signifies that Na-adsorbed graphene in 2 \(\times\) 2 supercell is less stable than its constituent systems (the pristine graphene and isolated Na atom), which agrees with the previously reported results for two-atom unit cell (Medeiros et al. 2010).

The supercells of larger size are more favorable for the adsorption of Na atom. Table 1 presents the binding energy values, the equilibrium distances from the average graphene sheet (h), the distance of adatom (Na) from the nearest carbon atom (\(d_{\rm AC}\)) of graphene and also the distortion of the graphene sheet \(d_{GC}\) in 3 \(\times\) 3 supercell. The table quantifies binding energies of sodium atom at the hollow (H), bridge (B), and top (T) sites as 0.45, 0.31 and 0.33 eV, respectively. The values imply that sodium atom is bound to all the tested sites of the graphene sheet and the H site is the most favored one among them. This conclusion agrees with the previously reported results (Chan et al. 2008; Oli et al. 2013). The equilibrium distance of sodium atom from the surface of graphene sheet is found to be 2.32 Å for H site, 2.42 Å for B site, and 2.41 Å for T site. The smallest equilibrium distance at H site signifies that there exists the strongest interaction between the sodium atom and carbon atoms of this site compared to those at B and T sites. The distortion produced by sodium is found only about 0.009 Å for hollow (H) site. The magnitude, however, is not significant (noticeable distortion is defined for \(\ge\)0.07 Å (Chan et al. 2008)) and this is true for bridge and top sites as well. Since the distortion in the graphene sheet (\(3\;\times\;3\) supercell) is not significant, C\(-\)C bonds near the Na atom retain their \(sp^{2}\) character and do not rehybridize significantly with any adsorbed atom orbitals.

Table 1 The adsorption energy of sodium atom \(\Delta\)E, its height from graphene sheet (h), the distance of the nearest carbon atom of the sheet (\(d_{AC}\)), and the distortion observed in \(3\;\times\;3\) graphene sheet due to adsorption of sodium atom (\(d_{GC}\)) are listed in this table
Full size table

In the case of \(4\;\times\;4\) supercell, the binding energy of sodium atom at hollow site is found to be 0.78 eV. This value is significantly higher than that for \(3\;\times\;3\) supercell at the same (H) site (0.45 eV). It has been found that the binding energy of adsorbed metal atoms is size dependent and gets almost constant value beyond certain number of C atoms in graphene. Oli et al. (2013) have reported rapid change in binding energy of alkali metal atoms (Li, Na and K) on graphene before reaching its saturation size, i.e., 48 number of carbon atoms. The quantities revealed by the authors for Na in passivated graphene structure of 16 carbon atoms of C\(_{16}\)H\(_{10}\) and 30 carbon atoms of C\(_{30}\)H\(_{14}\) are 0.32 eV and 0.72 eV, which are reasonably close to the present work of 18 (\(3\;\times\;3\) supercell) and 32 (\(4\;\times\;4\) supercell) atoms of graphene sheets, respectively (Oli et al. 2013). Ding et al. (2011), on the other hand, have reported different trends of size dependency of binding energy for different atoms, while varying graphene supercell from \(3\;\times\;3\) to \(4\;\times\;4\). The authors find binding energy difference in between \(3\;\times\;3\) and \(4\;\times\;4\) supercells for the elements of transition metals moving from Sc to Cu, in general, in decreasing trend. In separate calculations, we have observed very slight change in binding energy of Pt in \(4\;\times\;4\) graphene supercell with respect to that in \(3\;\times\;3\) graphene, while it is noticeable in between \(2\;\times\;2\) and \(3\;\times\;3\) (Lamichhane et al. 2014). It can, therefore, be concluded that size dependency of binding energy and also the saturation size of graphene depend on the chemistry of elements and usually higher for monovalent atoms.

The equilibrium distance between the sodium atom and graphene sheet in \(4\;\times\;4\) supercell, \(h=2.25\) Å, is lower than that in smaller supercells. Also the distance of the nearest carbon atom at hexagonal site from the sodium, \(d_{\rm AC}\), is found to be 2.69 Å. Both the geometrical parameters, h and \(d_{\rm AC}\), from the present results come in good agreement with the previous work performed by Chan et al. (2008) where the values are reported as 2.28 and 2.70 Å  respectively, in the same-size (\(4\times 4\)) supercell. Sodium atom produces a small distortion, 0.008 Å, to the graphene sheet of \(4\;\times\;4\) supercell as well. The distortion falls in the similar range to that of the 3\(\;\times\;\)3 supercell of graphene system with slightly higher value in the smaller size. The small positive distortion implies that graphene sheet gently moves towards the sodium atom (upwards), due to interaction between them.

The density of states (DOS) and the band structure calculations of 3\(\;\times\;\)3 pure and Na-adsorbed-graphene systems disclose the changes in electronic/magnetic properties of the systems, due to presence of Na, over the pure graphene.

Fig. 3
figure 3

Density of states (DOS) of pure graphene

Full size image
Fig. 4
figure 4

Density of states (DOS) of Na-adsorbed graphene

Full size image

The DOS for spin up and spin down states with reference to Fermi level (represented by the vertical dotted line) are shown in Figs. 3, and 4. In case of pure graphene (Fig. 3), the Fermi level lies at \(-\)2.35 eV and the DOS for spin up and spin down are seen symmetrical. Dirac point, where DOS is zero, lies exactly at the Fermi level. This symmetry of DOS about the Fermi level implies that pure graphene is non-magnetic. However, Dirac point shifts below the Fermi level and also the symmetry of DOS is broken with the extra peaks at/near the Fermi level when Na is adsorbed in graphene (Fig. 4). The shifting of Dirac point below the Fermi-level has also been discussed in Mukherjee and Kaloni (2012) for nitrogen-doped (N-doped) graphene system, where N contributes as an electron donor as similar to Na in the present work. Figure  4 also indicates its modified conductivity with reference to zero-band gap pristine graphene, which is consistent with the Medeiros et al. (2010). The breaking of symmetry, on the other hand, introduces the magnetic behavior. In case of 3\(\;\times\;\)3 supercell of graphene, the value of magnetization is 0.08 \(\mu _B\) where as that for 4\(\;\times\;\)4 supercell is 0.24 \(\mu _{\rm B}\). The total magnetization of 4\(\;\times\;\)4 supercell from this work agrees well with the previously reported value 0.27 \(\mu _{\rm B}\) (Chan et al. 2008). However, these values of magnetic moments for Na-graphene systems are lower than that for isolated sodium atom estimated theoretically (using electronic configurations) and computationally (present work) (1.0 \(\mu _{\rm B}\) ). We believe that this reduction in magnetization could be due to partial transfer of electronic charge from Na to the graphene.

Fig. 5
figure 5

The band structure of the pure graphene

Full size image
Fig. 6
figure 6

The bands structure of Na-graphene system. Thered arrow shows the s-orbital contribution of Na atom

Full size image

We have performed the band structure calculations by defining high symmetry points (i.e., K, \(\Gamma\) and M) on the edges of irreducible Brillouin zone in reciprocal space. In this case, we have taken 100 k-points along the specific direction of irreducible Brillouin zone to obtain the fine band structure. Figures 5 and 6 represent the band structure of pure graphene and Na-graphene systems, respectively.

Graphene is a single atomic layer of graphite where the carbon atoms condense in a honeycomb lattice due to their sp\(^2\) hybridization of \(2s\), \(2p_x\), and \(2p_y\) orbitals of three valence electrons out of four valence electrons. The mixing of the 2s and two 2p orbitals gives the planar sp\(^2\) hybridization. The electrons involved in sp\(^2\) hybridization are localized and form \(\sigma\) bonds. These electrons do not contribute to the transport phenomenon. The fourth electron, perpendicular to the plane of graphene sheet, occupies the \(2p_z\) orbital and forms the \(\pi\)-bond with \(2p_z\) electron of neighboring carbon atom (Novoselov et al. 2005). These electrons forming \(\pi\)-bond, however, are delocalized over the entire lattice and have higher energy than the electrons that form the \(\sigma\)-bonds. Most of the interesting properties of graphene arise due to these delocalized \(\pi\)-electrons (Novoselov et al. 2007). The bands above and below the Fermi level are referred to \(\pi ^{*}\) and \(\pi\) bands, respectively. In case of pure graphene, these conical \(\pi\) bands meet at the Fermi level and the band gap formed by them becomes zero (Fig. 5). The pure graphene thus is also known as zero band gap semiconductor. The meeting point of the conical bands (also known as Dirac point) and the conical region around the Dirac point (also known as Dirac cone) are the special properties of graphene and they are successfully produced in our calculations.

The band structure of Na-graphene system, on the other hands, shows that Dirac point shifts below the Fermi level by 1.084 eV (Fig. 6), similar to the results described by Liu et al. (2011). When the electron donor Na atom is adsorbed on graphene, a fraction of electronic charge transfers from the sodium atom towards the carbons of graphene. Since the \(\pi\) band of the graphene is already occupied, the transferred charge from sodium moves towards the \(\pi ^*\) band (Medeiros et al. 2010). If we compare the Fermi level of pure graphene and Na-adatom graphene, it lies at \(-\)2.346 eV in the former case and rises to \(-\)0.695 eV after the adsorption of Na atom. It means that the Fermi level shifts by 1.651 eV towards the conduction band (upwards) due to Na contribution. Furthermore, an extra band nearby Fermi level is clearly seen and marked as \(s\) orbital contribution of Na in Fig. 6, which is similar to Li contribution on graphene reported in Kaloni et al. (2013). The conduction band and valence band overlap in sodium-added graphene, and this suggests that the new material is conductor.

Adsorption of H\(_2\)-molecular/s

In this section, we discuss the geometry and the stability of adsorption of H\(_2\)-molecular/s on pure and sodium-adsorbed graphene systems.

Adsorption of H\(_2\)-molecular on the pure graphene

We have considered a number of samples to study the interaction between pure graphene sheet and single H\(_2\)-molecular in perpendicular and parallel orientations at different positions of 3\(\;\times\;\)3 supercell of graphene. The perpendicular configurations, H\(_2\) molecule perpendicular to the graphene plane at top, bridge and hollow sites, are shown in Figs. 7, 8, and 9, respectively. Figure 10, on the other hand, defines the parallel configuration of H\(_2\) at hollow position of the graphene.

Fig. 7
figure 7

H\(_2\) molecule perpendicular to the carbon atom (top site)

Full size image
Fig. 8
figure 8

H\(_2\) molecule perpendicular to the carbon–carbon bond (bridge site)

Full size image
Fig. 9
figure 9

H\(_2\) molecule perpendicular to the center of hexagonal ring of carbon atoms (hollow site)

Full size image
Fig. 10
figure 10

H\(_2\) molecule parallel to the hexagonal ring of carbon atoms

Full size image

The adsorption/binding energy (\(\Delta E\)) of H\(_2\) molecule, will be referred as adsorption energy hereby, is calculated from the optimized geometry of H\(_2\)-graphene system using equation

$$\begin{aligned} \Delta E = E_{G}+E_{H_2}-E_{G+H_2} \end{aligned}$$
(2)

with appropriate replacements of combined and individual components of the system. In this case, \(E_G\), \(E_{H_2}\), and \(E_{G+H_2}\) are the ground state energy of pure graphene, H\(_2\)-molecular, and \({H_2}\)-graphene (combined) systems, respectively. The values are calculated using same-sized supercell of graphene. The magnitudes of adsorption energy of H\(_2\) molecule and its distance from the graphene plane are presented in Table 2.

Table 2 The table quantifies the adsorption energy of single H\(_2\) molecule when adsorbed in perpendicular and parallel configurations at bridge (B), top (T) and hollow (H) positions of the graphene sheets (\(\Delta\)E). The symbol ‘h’ represents the equilibrium distance of H\(_2\) molecular, measured from the lower atom of \({H_2}\) in perpendicular case, from the graphene sheet
Full size table

Table 2 shows that the adsorption energy is maximum and the equilibrium distance is minimum for H\(_2\) molecular of its perpendicular configuration at hollow (H) site of graphene. This confirms that H\(_2\) molecular perpendicular to the hexagonal ring is most stable than any other configurations. The calculated values of adsorption energy of H\(_2\) molecular from the present work (61–69 meV) in the pure graphene sheet lie within the range of previously reported values (55–80 meV) (Kwon 2010). Small adsorption energy for single hydrogen adsorption could be because of very strong intra-molecular H–H bond, extremely small induced dipole moment, and the chemically inert nature of the our system (i.e., graphene). The small adsorption causes the weak binding of H\(_2\) mainly due to van der Waals interaction, and can be easily taken out from system, even at low temperature.

Adsorption of H\(_2\)-molecular/s on the sodium-adsorbed graphene

The adsorption of varying number of H\(_2\)-moleculars (one to five) has been tested on sodium-adsorbed 3\(\;\times\;\)3 graphene supercell, and the adsorption energy of the same is computed by

$$\Delta E = E_{{{\rm {G + Na}}}} + NE_{{{\rm {H}}_{{\rm {2}}} }} - E_{{{\rm {G + Na + H}}_{{\rm {2}}} }}$$
(3)

where \(E_{{{\rm {G + Na + H}}_{{\rm {2}}} }}\) is the ground state energy of the graphene supercell after adsorbing sodium atom and H\(_2\)-moleculars. Similarly, \(E_{{{\text{G + Na}}}}\) and \(E_{{{\text{H}}_{{\text{2}}} }}\) are the ground state energies of sodium-adsorbed graphene and molecular hydrogen, respectively. The number of H\(_2\)-moleculars is represented by N in the equation. The positive adsorption energy signifies that H\(_2\)-molecular bounds to the adatom-graphene system. The binding energy per H\(_2\)-molecular has been further calculated by,

B.E. per H\(_2\)-molecular = \(\frac{\Delta E}{N}\)

Figure 11 shows the optimized structures of hydrogen–sodium–graphene systems with a varying number of H\(_2\)-molecular.

Fig. 11
figure 11

Adsorption of a varying number of H\(_2\)-molecular on Na-adsorbed graphene system

Full size image
Table 3 The adsorption energy of H\(_2\)-moleculars (\(E_{\rm NH_2}\)) and adsorption energy per H\(_2\)-molecular (\(E_{\rm H_2}\)) are listed
Full size table

Table 3 shows that the adsorption energy of H\(_2\) molecular as well as adsorption energy per H\(_2\)-molecular increases continuously as we increase the number of molecules in the system. If we compare the adsorption energy per H\(_2\) molecular with and without adatom in graphene supercell, it can be found that presence of Na enhances the binding strength except for N = 1. The adsorption energy per H\(_2\) molecular ranges from 0.023 to 0.192 eV/H\(_2\) for the adsorption of one to five numbers of H\(_2\) molecular (Fig. 12). This range of adsorption energy comes in good agreement with the previously reported average binding energy of hydrogen, 0.09 to 0.22 eV for one to four molecules, in the Lithium-adsorbed graphene system (Ataca et al. 2008).

Fig. 12
figure 12

The binding energy per H\(_2\)-molecular is in increasing order up to 5 H\(_2\) molecular

Full size image

Figure 12 shows that the adsorption energy per H\(_2\)-molecular increases and attains the maximum value (0.192 eV) as we increase the number of H\(_2\)-molecular from one to five. The maximum value (0.192 eV/H\(_2\)) approaches nearby one of the goals set by DOE, 0.2–0.7 eV for the practical applications (Zhou and Willians 2011) . However, this increasing pattern is seen dissimilar with our calculations of hydrogen adsorption in other metallic atom adsorbed-graphene (Pd/Pt-graphene) systems. The reason could be because of different structural/electronic structure of sodium compared to other atoms. We believe that further analysis of charge distribution before and after the adsorption of gaseous molecules will explore the causes beyond these results.

Graphene and carbon nanomaterials have been considered as porous materials/substrates for the storage of energy carrying gases via adsorption at operating conditions. Similar to the previously accepted results, our calculations have found low adsorption energy of H\(_2\) in pure graphene. However, the binding strength is enhanced significantly for 2–5 numbers of H\(_2\)-molecular in Na-adsorbed graphene. The content of hydrogen in the proposed system can also be analyzed on the basis of its presence by weight. For 5 molecular of hydrogen, the maximum number of molecules adsorbed in our calculations reaches 4.02 wt % (weight percentage) which is progressive towards the DOE (Department of Energy) target (around 6 wt %) for the practical applications (Durgun et al. 2008).

Conclusions

The structural and electronic properties of pure graphene and Na-graphene systems are studied. The binding energy values for Na at different sites suggest that the hollow site is the most favorable position for the adsorption of sodium atom on graphene sheet. The band structure and DOS calculations of the pure graphene find zero band gap and non-magnetic nature, which are consistent with the previously established facts. In case of sodium-adsorbed graphene system, the conduction band and valence band are overlapped. DOS are not symmetrical, which indicates that the system is magnetic, and the calculations find magnetic moment 0.24 \(\mu _{\rm B}\) for Na in \(4\;\times\;4\) graphene supercell.

We have also performed the adsorption of H\(_2\)-molecular on pure graphene sheet and sodium-adsorbed graphene system. The present study reveals that the adsorption energy per H\(_2\)-molecular is enhanced due to the presence of Na atom and reaches 0.192 eV in case of Na-adsorbed graphene compared to its value 0.068 eV in pure graphene. The hydrogen storage capacity of Na-adsorbed graphene for the adsorption of five H\(_2\)-molecular has been calculated as 4.02 wt % (weight percentage), which is progressive towards the DOE (Department of Energy) target (around 6 wt %) for the practical applications.