Introduction

Carbon is an important element known from ancient times and exists in a variety of forms like diamond, graphite, conducting and semiconducting nanotubes and fullerenes. In the recent past closed cage fullerenes have attracted researchers due to their unique structures and properties. The fullerene molecule is having a structure of regular truncated icosahedron. It has 60 carbon atoms located at the vertices of a truncated icosahedron. The icosahedron structure has 90 edges of same length, 60 identical vertices, 20 hexagonal faces and 12 pentagonal faces to form a closed cage (Dresselhaus and Eklund PC 1996). The boundary between a hexagon and a pentagon is a single carbon–carbon bond of bond length 1.41 Ǻ and between two hexagons is a double bond of bond length 1.46 Ǻ. Because of the large size of fullerene molecule as compared to alkali metal dopant atoms or ions, the inner cavity in a C60 molecule can accommodate numerous guest species.

Various concepts have been developed to manipulate, design, characterize and to understand the potential of fullerene as a building block for stable electron donor–acceptor hybrids and conjugates (Nierengarten et al. 2003; Guldi 2003; El-Khouly et al. 2004; Guldi et al. 2004; Segura et al. 2005). Doping is a very famous and effective method for changing the properties of many materials. So Fullerene has been extensively doped (Basiuk et al. 2012; Celasu 2013) to tailor its properties. When fullerene is doped with different atoms, it can change the properties of the system by enhancing their chemical reactivity but the stability of structures is maintained. Exohedral-doped fullerene has shown superconductivity in fullerides (Hebard et al. 1991). Endohedral metallofullerenes can accommodate additional atoms in them (Akasaka et al. 2010). These fullerenes form interesting complexes that are formed due to electron transfer between the fullerene cage and the inner species. Endohedral metallofullerenes are more interesting due to their significant control over the chemical and physical properties, which can be obtained by just changing the nature and composition of doped species (Guldi et al. 2009; Rudolf et al. 2012; Lu et al. 2011). Modeling techniques help in a better way to understand the physical properties of fullerenes and its derivatives. In the literature ab initio calculations have been used to study the structural and electronic properties of small silicon clusters as well as endohedral metallofullerenes (Andreoni 1993; Ibrahim et al. 2006). Ab initio calculations on the stability of MC60 (M = Sc, Y and K) and Si atom-doped endohedral C60 have been reported in the literature (Misra et al. 2012: Suzuki et al. 1994; Ibrahimand El-Haes 2005; Lu et al. 2001). Furthermore, doping of radioactive elements, such as beryllium (Be) (Ohtsuki et al. 1996) and gadolinium (Gd) (Kubozono et al. 1996), has been found to be properly encaged inside a C60 cluster, leading to substantial applications in nuclear wasting disposal such as nano-container radio activities. Indeed, the discovery of fullerenes has opened the gate for the production and investigation of much more suitable material building blocks, which could be used as drug deliverers in medical applications and/or possible superconductors in technological applications (Alzahrani2012). Therefore, fullerene provides an entirely new branch of chemistry, materials science, and physics. Moreover, solid fullerene is treated as a semiconductor material (Saito and Oshiyama 1991).

Large clusters of Cu, Ag, Au, Al and Ga have been extensively studied using tight binding model (Cleri and Rosato 1993). Calculation of cohesive energy and bulk properties of alkali-doped C60 solids with 6-exponential potential is reported (Kumar et al. 2005). Structural and thermodynamic properties of dimer C60 fullerene solid are extensively studied with 6-exponential potential (Kaur et al. 2000). Structural properties of armchair single-wall carbon nanotubes under hydrostatic pressure is reported using Tersoff potential (Imtani and Jindal 2005). Endohedral doped C60 with Si, Ge, Au and Tl has been studied with DFT study and model potentials by our group (Umran and Kumar 2014; Umran et al. 2015). In the present work, we have studied structural properties of C60 fullerene molecule doped with Cu, Ag, Al and Ga atoms using model potentials and compared the results of BE/dopant atom with DFT study done with SIESTA (Spanish Initiative for Electronic Simulations with Thousands of Atoms) code.

Model potentials

In case of metal-doped fullerene total energy may be written in terms of three different types of interactions, namely carbon–carbon interaction, metal–metal interaction and carbon–metal interaction. Brenner potential, Gupta potential, and Lenard Jones potentials have been used to model these interactions, respectively. Generally parameters used in these model potentials are not readily available and need to be fine-tuned for different dopants. Brenner potential (Brenner 1990) has been used to study properties of several carbon based systems like graphite, diamond, fullerenes and carbon nanotubes (Gupta et al. 2005; Robertson et al. 1994). The interaction between carbon and metal is calculated by Lennard–Jones potential (Guan et al. 1996), the interaction between metal–metal is calculated by Gupta potential (Gupta 1981). These potentials have been used for calculating the ground state energies of endohedral fullerenes (Umran et al. 2015). Total energy of the metal-doped endohedral fullerene may be written as

$$ E = \frac{1}{2}\mathop \sum \limits_{{\begin{array}{*{20}c} {i,j} \\ {i \ne j} \\ {C - C} \\ \end{array} }} V_{i,j}^{B} + \frac{1}{2}\mathop \sum \limits_{{\begin{array}{*{20}c} {i,j} \\ {C - M} \\ \end{array} }} V_{i,j}^{L - J} + V_{M - M}^{G} , $$
(1)

where \( V_{\text{i,j}}^{\text{B}} \) (Brenner potential) corresponds to the energy between carbon–carbon atoms. \( V{}_{\text{i,j}}^{\text{L - J}} \) (Lennard–Jones potential) is the energy between carbon–metal atoms and \( V_{\text{M - M}}^{\text{G}} \) (Gupta potential) is the energy between metal–metal atoms. Total energy of the pristine fullerene may be expressed as

$$ E_{C60} = \frac{1}{2}\mathop \sum \limits_{{\begin{array}{*{20}c} {i,j} \\ {i \ne j} \\ {C - C} \\ \end{array} }} V_{i,j}^{B} $$
(2)

Binding energy/dopant becomes \( BE = \frac{{E - E_{C60} }}{No. of dopants} \). The potentials used are briefly described in the following section.

Brenner potential

Brenner potential is a class of empirical potentials used to model covalently bound systems. Thus, Brenner potential is able to describe the physical properties for compound interactions correctly. The potential energy between any two carbon atoms on C60, say i and j, separated by a distance r ij is given as:

$$ V_{ij}^{B} = V_{R} \left( {r_{ij} } \right) - B_{ij} V_{A} \left( {r_{ij}} \right), $$
(3)

where

$$ V_{R} \left( {r_{ij} } \right) = f\left( {r_{ij} } \right)\left[ {\frac{{D_{ij}^{{\left( {e} \right)}} }}{{S_{ij} - 1}}} \right]{ \exp }\left[ { - \beta_{ij} \sqrt {2S_{ij} } \left( {{r} - {R}_{ij}^{{\left( {e} \right)}} } \right)} \right] $$
(4)
$$ V_{{A}} \left( {r_{{ij}} } \right) = f\left( {r_{{ij}} } \right)\left[ {\frac{{D_{{ij}}^{{\left( {{e}} \right)}} S_{{ij}} }}{{S_{{ij}} - 1}}} \right]{ \exp }\left( { - \beta_{{ij}} \sqrt {\frac{2}{{S_{{ij}} }}} \left( {r - R_{{ij}}^{{\left( {{e}} \right)}} } \right)} \right) $$
(5)
$$ f\left( {r_{ij} } \right) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {if\; r_{ij} < R_{ij}^{(1)} } \hfill \\ {\frac{1}{2}\left[ {1 + cos\left( {\frac{{\pi \left( {r_{ij} - R_{ij}^{(1)} } \right)}}{{R_{ij}^{(2)} - R_{ij}^{(1)} }}} \right)} \right] ,} \hfill & { if\;R_{ij}^{(1)} < r_{ij} < R_{ij}^{\left( 2 \right)} } \hfill \\ {0,} \hfill & {if\; r_{ij} > R_{ij}^{(2)} } \hfill \\ \end{array} } \right. $$
(6)

Here, V R and V A are the repulsive and attractive potential energy terms, respectively. \( R_{{ij}}^{{ ( {{e)}}}} \) is the equilibrium distance between atoms i and j. \( S_{{ij, }} D_{{ij}}^{{\left( {{e}} \right)}} , \beta_{{ij}} \) are the constants that reflects the types of atoms i and j. The screening function f(rij) restricts the interaction to nearest neighbors as defined by the values of \( R_{{ij}}^{ ( 1 )} \) and \( R_{{ij}}^{ ( 2 )} \). In addition, the Brenner potential takes bonding topology into account with empirical bond order function B ij. Table 1 shows the parameter set for the Brenner empirical potential (Brenner 1990).

Table 1 The parameters of Brenner Potential (Brenner 1990)
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Lennard–Jones potential

Lennard–Jones potential is employed to model the van der Waals effect in molecular collision simulations. The form of the Lennard–Jones potential is very similar to.

$$ V_{\text{ij}}^{{{\text{L}} - {\text{J}}}} = 4\varepsilon \left[ {\left( {\frac{\sigma }{{r_{ij} }}} \right)^{12} - \left( {\frac{\sigma }{{r_{ij} }}} \right)^{6} } \right] $$
(7)

Here \( \sigma \) corresponds to the distance between two atoms at equilibrium and ɛ represents the strength of interaction. The optimized parameters (ε, σ) for Cu–C, Ag–C, Al–C and Ga–C used in present calculations are listed in Table 2.

Table 2 Lennard–Jones potential parameters
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Gupta potential

The many-body Gupta potential is used to study the structure and properties of metal–metal clusters. Gupta potential is derived from Gupta’s expression (Gupta 1981) for the cohesive energy of bulk material. It includes a repulsive pair \( V_{\text{ij}}^{\text{r}} \) and many-body attractive terms \( V_{{_{\text{ij}} }}^{\text{m}} \).

$$ V^{{G}} = \mathop \sum \limits_{{{\text{i}} = 1}}^{\text{N}} \left[ {\mathop \sum \limits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne i} \\ \end{array} }}^{\text{N}} V_{{ij}}^{{r}} \left( {{\text{r}}_{{ij}} } \right) - \sqrt {\mathop \sum \limits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne i} \\ \end{array} }}^{\text{N}} V_{{ij}}^{{m}} } } \right], $$
(8)
$$ V_{\text{ij }}^{\text{r}} = {\text{A exp}}( - p_{{ij}} \left[ {\frac{{{\text{r}}_{{ij}} }}{{{\text{r}}_{0} }} - 1} \right], $$
(9)
$$ V_{ij }^{m} = \zeta^{2} \exp \left( { - 2q_{ij} \left[ {\frac{{r_{ij} }}{{r_{0} }} - 1} \right]} \right), $$
(10)

where r ij is the distance between two atoms \( A, r_{0} , \xi , p, q \) are the parameters fitted to experimental values of the cohesive energy, lattice parameters and independent of elastic constants for the reference crystal at 0 K. The Gupta potential parameters used in present calculations are listed in the Table 3.

Table 3 Gupta potential parameters
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Results and discussion

In this part, we show the optimized structures of Mn@C60 (M = Cu, Ag, Al and Ga) molecules. Initial input coordinates of pure C60 fullerene molecule and doped fullerenes have been taken from DFT calculations (Dhiman et al. 2015). Then the calculations were done using model potentials as discussed in “Model potentials” section. The parameters for model potential calculations have been taken from the literature and the calculations have been done for structural analysis and BE/dopant atom. We observed that the result for BE/dopant atom differ much from the result obtained from DFT study calculations. Then we tune few parameters like \( R_{\text{ij}}^{ ( 2 )} \varvec{ } \) and \( r_{o} \) as shown in Table 2 and Table 3. The results obtained with new parameters are in excellent agreement with DFT calculations. The calculated values of single and double bonds in fullerene molecule are between 1.41 and 1.46 Å, respectively. These values are in good agreement with the experimental values between 1.40 and 1.46 Ǻ (Hedberg et al. 1991). First, we focus on the study of structural properties of copper and silver doped fullerenes and then aluminum- and gallium-doped fullerenes have been discussed.

Cu and Ag atoms doped inside C60 (Group II elements)

The optimized fullerene structure is used to dope copper and silver atoms. First of all, initial coordinates are assigned to copper and carbon atoms of C60 molecule and the system was allowed to relax without additional constraints. The ground state structures of copper-doped fullerenes are shown in Fig. 1. In Cu1@C60 complex single copper atom stabilizes in the center of C60 (Fig. 1a). In Cu2@C60 structure two Cu atoms exist in a dimer form having Cu–Cu bond distance is 2.22 Å inside the fullerene cage. The dimer lies in the center of two hexagons of the fullerene ring (Fig. 1b). Bond length of dimer as calculated by DFT study is 2.16 Å. There is only a difference of 0.06 Å between the two calculations. The carbon–copper bond distance is 2.59 Å. In case of Cu3@C60, Cu3 exist in triangular structure (Fig. 1c) with copper–copper bond distance 2.23 Å and carbon–copper bond distance varies from 2.65 to 2.69 Å. In Cu4@C60, Cu4 shows a tetrahedron structure (Fig. 1d) with copper–copper and carbon–copper bond lengths of 2.21 and 2.63 Å respectively. InCu5@C60, cluster of five copper atoms shows a stable triangular bipyramidal structure lies in the center of C60 with copper–copper and carbon–copper bond lengths 2.3, 2.1 and 2.47–2.58 Å respectively as presented in (Fig. 1e). Cluster of Cu6 in Cu6@C60 exist in a square bipyramid (Fig. 1f) form lies symmetrically in the center of two hexagons with copper–copper and carbon–copper bond lengths 2.13, 2.15 and 2.43–2.64 Å. Cu7@C60 form a stable structure with Cu7 has a perfect pentagonal bipyramidal structure having copper–copper and carbon–copper bond lengths 2.04, 2.18, 2.40–2.67 Å respectively (Fig. 1g). In Cu8@C60 complex (Fig. 1h), Cu8 forms a distorted cube of copper–copper bond length varies from 2.03–2.06Å and all copper atoms form bonds with carbon atoms of the cage having carbon–copper bond length varies from 2.37–2.63Å and in Cu9@C60 (Fig. 1i), Cu9shows a body centered cube with copper–copper bond length 2.17 Å, the body centered copper atom is at a distance of 1.93 Å from other two copper atoms situated at the diagonal and carbon–copper bond length varies from 2.37–2.56 Å, respectively. Doping of 10th copper atom in the fullerene molecule exerts a strain in carbon–carbon bond and the cage breaks. As the size of copper clusters inside the fullerene increases, distortion of cage increases as it cannot withstand the pressure exerted by cluster atoms.

Fig. 1
figure 1

Ground state structures of Cun@C60 (n = 1−9)

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In the same way, we have done doping of Ag atoms in the C60 cage and the ground state structures of Agn@C60 (n = 1−9) are presented in Fig. 2. On doping of single Ag atom in the fullerene molecule produce Ag1@C60 complex. Ag atom is placed at different positions inside the cage and found that is stable only in the center of the C60 molecule. Encapsulation of two Ag atoms inside the fullerene cage leads to the formation of Ag2@C60 complex. In this case Ag2 dimer is stable like Cu2 dimer and lies symmetrically in the center of two hexagons of the cage. Ag2 molecule does not make any bond with carbon atom of the fullerene.

Fig. 2
figure 2

Ground state structures of of Agn@C60 (n = 1−9)

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The calculated Ag–Ag interatomic distance is 2.38 Å, which is 0.17 Å less than bare optimized Ag2 dimer. In Ag3@C60, Ag3 has a triangular structure similar to Cu3@C60. The bond length of Ag–Ag dimer is 2.41 Å and this structure does not form any bond with carbon atom of the cage. In the case of Ag4@C60, Ag4 has a tetrahedron structure with silver–silver bond length 2.45 Å and all the four Ag atoms form bond with carbon atoms of the cage with carbon–silver bond length 2.70 Å. Ag5 in Ag5@C60 exist as a trigonal bipyramid structure with bond lengths 2.62 and 2.41 Å. All the five silver atoms form bonds with carbon atoms of the cage with bond length 2.65 Å. In Ag6@C60 complex, Ag6 cluster has a square bipyramid structure with silver–silver and carbon–silver bond lengths 2.46, 2.45 and 2.67 Å respectively. Inside Ag7@C60 complex, Ag7 cluster form a pentagon bipyramid structure, silver-silver and carbon–silver bond lengths are 2.33, 2.59–2.68 Å respectively. In Ag8@C60 complex, Ag8 form a distorted cube with Ag–Ag bond length varies from 2.33–2.41Å. All eight Ag atoms form bonds with carbon atoms of the fullerene cage and carbon–silver bond length varies from 2.59–2.71 Å.

On doping of cluster of nine silver atoms, the C60 cage breaks as shown in Fig. 2a. The variation of BE/dopant atom of Cun@C60 and Agn@C60 with number of dopant atoms as calculated with model potentials and with DFT study is plotted in Fig. 3. The results indicate that the reported parameters have reproduced DFT results (Dhiman et al. 2015) qualitatively as well as quantitatively. Therefore, the model of interactions and new parameters describes endohedral systems with great accuracy.

Fig. 3
figure 3

Variation of BE/Cu or Ag atom with number of atoms

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Doping of Al and Ga atoms inside C60 (group III elements)

The relaxed structures of Aln@C60 (n = 1−9) and Gan@C60 (n = 1−9) are presented in Figs. 4 and 5. The calculated Al–Al bond distance in relaxed structure is 2.59 Å. This is almost 0.1 Å less than the value reported (Wang et al. 2000) and agreed well with the experimental value 2.56 Å (Rosen1970). The Calculated Ga–Ga interatomic distance is found to be 2.62 Å, which is 0.06 Å less than the value reported (Balasubramanian 1990). In both cases it is found that the fullerene complexes are stable up to n = 1−9 with a small distortion of the cage due to large size of dopant clusters. But on addition of tenth aluminum or gallium atom inside the C60 cage, it breaks.

Fig. 4
figure 4

Ground state structures of Aln@C60 (n = 1−9)

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Fig. 5
figure 5

Ground state structures of Gan@C60 (n = 1−9)

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This is due to the fact that the empty space inside C60 is not sufficient to accommodate large clusters. It is also inferred that the stability of the fullerene cage generally depend upon the number of dopant atoms (Al, Ga) atoms inside the cage. BE/dopant atom (Al/Ga) of Aln@C60 and Gan@C60 (n = 1−10) with number of dopant atoms as calculated with model potentials and DFT study is presented in Fig. 6. We found that the BE/Al atom follow the same trend as we have found in DFT calculations.

Fig. 6
figure 6

Variation of BE/Al or Ga atom with number of atoms

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In case of Ga-doped fullerene, BE/Ga atom decreases up to four Ga atoms and then it becomes positive. This implies that complexes are in metastable state and still preserve the symmetry and cage breaks only after insertion of tenth gallium atom. So the results obtained with potential calculations are in good agreement with the DFT study and again establishes the authenticity of model potentials and new set of interaction parameters. From the structure analysis it is found that the stability of fullerene shell strongly depends upon the number of dopant atoms in the cage.

Conclusions

In the present paper, we have investigated the structures and BE/dopant of Cu, Ag, Al and Ga doped endohedral fullerenes with the help of model potentials. Brenner potential, Lennard–Jones potential and Gupta potential are used to study carbon–carbon, carbon–metal and metal–metal interactions. Total energy calculations have been done to find out the ground state energy structures and BE/dopant atom. In Cun@C60 molecule, maximum nine copper atoms can be doped inside the fullerene cage and form stable complexes without distorting the cage significantly. The addition of tenth copper atom inside the cage increases the size of the copper cluster and produces a large pressure on the wall of the cage. As a result, the strength of carbon–carbon bonds become weak and this leads to the breaking of fullerene cage. The space inside the fullerene cage is not sufficient to accommodate large size clusters beyond nine atoms. Similarly maximum eight silver atoms, nine aluminum atoms and nine gallium atoms can be doped inside the fullerene cage those can make stable complexes. The calculated values of BE/dopant atom with model potentials are in excellent agreement with the values calculated with DFT study. This validates our results of model potentials and new set of interaction parameters.