Studying clusters of various chemical elements has become a modern research topic in both physical and chemical communities over the last four decades^{[1, 2]}, due to the size-dependent evolution of their fundamental properties and their technological applications in large variety of research fields, from catalysis to optoelectronics. Their particular structural, energetic, and electronic properties are fully understood and still constitute the subject of many research projects^{[3, 4]}. Nanoscale materials (called clusters) with various sizes can provide different behaviors to that of the bulk material. The physical and chemical features of bimetallic clusters are dependent not only on the size and shape but also on the chemical composition and the atomic arrangement of the two metal elements^[2]. Therefore, studying the changes in the structural and electronic properties of the cluster with its size has become important^{[4, 5]}. Several theoretical calculations have been performed on pure and mixed neutral and charged clusters of group 14 elements^{[6, 7]} especially silicon and germanium.

Numerous theoretical and experimental studies on Ge clusters have been published over the last decade^[8] and different types of structures have been proposed^{[9, 10]}. Theoretically, several computations have concluded that Ge_n cages can be stabilized through the encapsulation of guest atom inside the cage. This was seen before in Si_n cages. Matthias Brack et al.^[11] presented the global minimum of CuGe₁₀⁺ clusters as a magic number along with D_4d symmetry. Han et al.^[12] have presented a theoretical investigation of very small Ge_n (n = 1–4) clusters doped by Sn, and they found a charge transfer from Sn to Ge atoms. Singh et al.^[13] have reported that the n capsulation can be utilized for stabilizing highly symmetric Ge_n cages having from 16 to 20 atoms. Wang and Han^[14] have investigated CuGe_n (n = 2–13) clusters and shown that Cu doping can decreaseits binding energies, and so, the stability of Ge_{n + 1} clusters. Zhao and Wang have studied in 2009 Mn-doped Ge_n clusters^[15] and shown that Mn dopant can contribute to the stability increase of Ge_n+1 clusters. Jaiswal and Kumarusing studied the atomic and electronic structures of both neutral and negatively charged ZrGe_n (n = 1–21) clusters using ab-initio calculations^[16] and predicted cage-like stable geometries for n ≥ 13. Siouani et al.^[10] have investigated systematically the equilibrium geometries and electronic properties of VGe_n (n = 1–19) clusters and found that V atom in VGe_n can make the stability stronger starting from n = 7. More recently, Mahtout et al.^[17] have studied the structural, energetic, and electronic properties of MGe_n clusters with M = Cu, Ag, Au and n = 1–19 using DFT approach. They have found the endohedral structures where the metal atom was incorporated inside the Ge_{n + 1} cage appear at n = 10 when the dopant is Cu and at n = 12 for Ag or Au. Djaadi et al.^[18] have investigated the structures and relative stability of pure Ge_{n + 1}, neutral cationic and anionic SnGe_n (n = 1–17) clusters. They found that the Sn atom occupied a peripheral position for SnGe_n clusters when n < 12 and occupied a core position for n > 12.

Here, we report a systematic computational study based on the density functional theory (DFT) aiming to highlight the possible effects of one arsenic atom on the structural, energetic, and electronic properties of different isomers of Ge_{n + 1} in the atomic size range n = 1–20 atoms. We believe this work is useful for deeply understanding the effects of incorporating one As atom into Ge_{n + 1} clusters and can be considered as a guideline for future experiments. To the best of our knowledge, no systematic study has been addressed on neutral and charged AsGe_n clusters.

The electronic structure calculations of AsGe_n^q (n = 1–20, q = 0, ± 1) clusters were performed using the density functional theory (DFT)^[19] as implemented in the SIESTA program^[20]. This code uses norm-conserving Troullier-Martins nonlocal pseudopotentials^{[2, 21]} and employs flexible basis sets of localized Gaussian-type atomic orbitals^[2]. The exchange correlation energy was evaluated using the generalized gradient approximation (GGA) parameterized by Perdew and Zunger^[22] and by Perdew, Burke, and Ernserh of (PBE)^[23]. The self-consistent field (SCF) calculations were carried out with convergence criterion of 1 × 10⁻⁴ a.u. for total energy. We used a double ζ (DZ) basis with polarization function for As and Ge atoms. With energy shift parameter of 50 meV, the change density was calculated in regular real-space grid with cut-off energy of 150 Ry. The simulated clusters were placed in a big cubic supercell with a parameter of 40 Å, including enough vacuums between neighboring clusters and periodic boundary conditions were imposed. To sample the Brillouin zone, only a single k-point centered at Γ was used because of the extended size of the supercell. The conjugated gradient method within Hellmann-Feynman forces was used and all the forces after structural relaxation were less than 10⁻³ eV/Å.

We first searched for the lowest-energy structures of pure Ge_{n + 1} clusters in the 1–20 atoms range by exploring various possibilities of isomers. Secondly, the most stable ground state structures obtained for Ge_{n + 1} clusters were doped through substitution with one As atom. Then, the obtained AsGe_n clusters were optimized until reaching their ground states. In order to get lowest-energy structures of the AsGe_n clusters, several initials isomeric structures, including some high and low symmetries, were optimized by placing one As atom in substitution in different possible sites of the pure corresponding Ge_{n + 1} in order to get as close as possible to the low energy structures. Then, we cannot be sure that a more stable structure than those found in our calculations does not exist. We aim of our study is to highlight the variation of the properties of germanium cage clusters due to the As doping atom. We hope that this work would be useful to understand the influence of the As atom on the properties of germanium clusters and provide some guidelines for the probable future experimental studies. To check the validity of our computational method, benchmark tests have been done on Ge₂, Ge₃, and As₂ parameters. The values are reported in Table 1 together with available theoretical and experimental results. Our calculated results were found to be in line with the literature, confirming the reliability of our protocol to simulate small Ge clusters.

We report in Fig. 1 the lowest-energy structures obtained for Ge_{n + 1} (n = 1–20) and their corresponding isomers. Their energetic ordering is reported in Table 2. Our calculations reveal that almost all atoms are on the surface. Until n = 20, prolate-type geometries are in competition with the nearly spherical ones. The calculated results for the most favorable isomers are given in bold character. The most stable structures for n + 1 = 2, 3, and 4 adopt a planar disposition in line with previous works^{[10, 14, 18, 24, 25]} using DFT different calculations. The triangular geometry with C_2v symmetry is found to be the lowest-energy structure for Ge₃. The lowest-energy isomer of the tetramer Ge₄ has D_2h symmetry in line with previous findings^{[15, 18, 20, 26, 27]}. The most favorable isomer for Ge₅ cluster reveals a triangular bipyramid disposition with D_3h symmetry, which is also in line with the previous data^{[10, 14, 24, 26]}. The lowest-energy Ge₆ cluster has bicapped quadrilateral structure with C_2v symmetry, in good agreement with the previous data^{[10, 18, 26]}. For Ge₇ cluster, the most stable structure reveals a pentagonal bipyramid of D_5h symmetry. Other researchers also previously obtained similar results for Ge₇^{[10, 14, 24, 26]}. For Ge₈ clusters, the most stable isomer shows a capped pentagonal bipyramid disposition of C_2v symmetry, as obtained in earlier works^{[20, 28]}. Ge₉ is a capped pentagonal bipyramid structure of C_2v symmetry. The most favorable isomer of Ge₁₀ cluster is a capped pentagonal basis structure and has C_3v symmetry. The lowest-energy Ge₁₁ cluster shows a compact near spherical geometry of C_s symmetry. As per Ge₁₂ and Ge₁₃ clusters, prolate-type structure with C_2v symmetry was always preferred. The shape of Ge₁₄ is a prolate structure with C_s symmetry. For Ge₁₅ and Ge₁₆ clusters, prolate-type geometry was obtained with C_2v and C_2h symmetries, respectively. The lowest-energy isomer for Ge₁₇ possesses a near spherical geometry of C_s symmetry. From Ge₁₈ to Ge₂₁ clusters, prolate-type shape with C₁ symmetry was always preferred.

The most favorable geometries of AsGe_n (n = 1–20) clusters and their corresponding isomers are summarized in Fig. 2, whereas their energetic ordering is reported in Table 3. The AsGe_n clusters adopt somehow similar structures to their corresponding Ge_{n + 1} except for n = 8, 10, 11, and 16. In all cases, the arsenic atom is always located on the surface. The AsGe₂ cluster shows a triangular geometry of C_2v symmetry with two equivalent As–Ge bonds of 2.445 Å and one Ge–Ge bond of 2.775 Å. The As–Ge bond distance of 0.09 Å is larger than that in AsGe dimer. The most stable structure of AsGe₃ cluster presents a planar C_2v symmetry with a binding energy of 2.418 eV/atom, which is smaller than that of tetramer Ge₄ (2.557 eV/atom). For AsGe₄, a distorted rectangular pyramid with C_2v symmetry is found with a binding energy of 0.066 eV/atom, which is also smaller thanGe₅. The Ge–Ge and As–Ge bond lengths are 2.692 and 2.734 Å, respectively. For AsGe₅, the As atom is located at the convex site of a quasi-rectangular bipyramid structure of C_4v symmetry, As–Ge bond distance of 2.679 Å, and an average Ge–Ge bond distance of 2.807 Å. The lowest energy isomer for AsGe₆ cluster is a structure with C_2v point group symmetry, As–Ge bond length of 2.704 Å, and an average Ge–Ge bond distance of 2.786 Å. For AsGe₇ cluster, the lowest-energy isomer reveals a low-lying structure with a planar C_3v symmetry and a binding energy of 2.835 eV/atom, which is smaller than that for tetramer Ge₈ (2.866 eV/atom). For AsGe₈ cluster, its binding energy of only 0.076 eV/atom is also smaller than that obtained for Ge₉ cluster with C_s symmetry of the ground state isomer. The lowest-energy structure of AsGe₉ cluster has C_s symmetry combining two irregular hexagonal prisms with As atom on top of one of them. The ground state geometry of AsGe₁₀ has C₁ point group symmetry. The As atom tends to be stabilized on the surface. For n = 11, 12, 13, 14, 15, 16, and 17, prolate structures were found to be the most stable in their ground state. Its binding energies are much smaller than Ge_{n + 1}. AsGe₁₈ has a lowest-energy structure with C₁ point group symmetry. The As atom tends to be stabilized on the surface. The most favorable isomer for AsGe₁₉ cluster shows prolate-like and cage-like structures with C₁ symmetry and a calculated binding energy of 3.029 eV/atom, which is close to that of tetramer Ge₂₀ (3.046 eV/atom). For n = 20, the lowest-energy isomer combines a prolate-like structure with the cage-like one. The binding energy of AsGe₂₀ (0.004 eV/atom) is almost the same than that obtained for the ground state structure of the pure Ge₂₁ cluster.

The size dependence on the binding energies per atom for the lowest energy structures of Ge_{n + 1} and AsGe_n (n = 1–20) clusters are shown in Fig. 3. As expected, the bonding energy gradually increases with increasing size, and this can be associated with the increasing average number of neighbors per atom. For AsGe_n, we observe that the binding energies are lower than those for Ge_{n + 1}. This means that doping with As atoms has no immediate effects on enhancing the stability of germanium cluster at small size. In most of AsGe_n clusters, the final structures do not differ from that of the corresponding pure germanium cluster. This may be due to the equivalence in the nature of bonding, the size and the atomic mass between the two metalloids arsenic and germanium used in this study. However, for n = 2 and n = 20 we observe that the binding energy per atom of AsGe_n clusters is larger than those of corresponding pure Ge_{n + 1} clusters. Then, the substituting a Ge atom by a As atom increases the stability these two clusters. An increase in the binding energy is obtained with 1.426 eV for n = 2 to 2.837 eV for n = 6, and then non-monotonic and slow growth could be reached until n = 20.

Fig. 4 shows the plot of the size dependence on the fragmentation energies of Ge_{n + 1} and AsGe_n (n = 1–20) clusters. An oscillating behavior is observed. The clusters with large values of fragmentation energy are relatively stronger in thermodynamic stability than neighboring clusters. Consequently, the thermodynamic stabilities of Ge₅, Ge₈, Ge₁₀, Ge₁₁, AsGe₆, AsGe₉, AsGe₁₂, and AsGe₂₀ clusters are relatively strong.

The evolution of the second-order difference of energies for the most favorable structures of Ge_{n + 1} and AsGe_n (n = 1–20) clusters as a function of the cluster size is plotted in Fig. 5. The curve shows pronounced peaks for AsGe_n at range size n = 3, 5, 7, 10, 11, 13, and 19 atoms. This suggests these clusters to be more favorable than their neighbors. . In cluster physics, if the values of Δ₂E are positive this means that the dissociation of As atom is an unfavorable process and the clusters are particularly stable. It can also be seen that the curve of Ge_{n + 1} clusters with range size n = 2, 3, 5, 6, 8, 11, 12, 13, 15, and 20 exhibit higher stability than their neighbors. As a consequence, the stability of AsGe_n structures with n = 3, 5, 11, 13 atoms correlates with the stability of the corresponding Ge_{n + 1} structures, where the AsGe_n structure was maintained the same upon the incorporation of As dopant.

In order to obtain insight into the kinetic stability of AsGe_n clusters, we calculated and analyzed the HOMO–LUMO gap. In general, the reactivity of the cluster decreases with increasing the HOMO–LUMO gap^[28]. Fig. 6 reports the size dependence of HOMO–LUMO gap for the most favorable structures of Ge_{n + 1} and AsGe_n (n = 1–20) clusters. The decreasing behavior with the size is important for Ge_{n + 1} clusters, while is less pronounced for AsGe_n clusters. Overall, the gaps of AsGe_n are much lower than those obtained for Ge_{n + 1} clusters, except for n = 3 and 20, The value for AsGe_n roughly oscillates in between 0.171 and 1.278 eV, which indicates a chemical activity increase of Ge_{n + 1} clusters when doped with As. Doping Ge_{n + 1} cages with an As atom leads to a significant HOMO–LUMO gap reduction in AsGe_n clusters. This means that the chemical activity of AsGe_n is higher than that of Ge_{n + 1} clusters and the inserted As atom highlights the metallic character of AsGe_n clusters. It should also be noted that Ge₄ cluster possesses the largest HOMO–LUMO gap of 2.036 eV, which indicates that Ge₄ cluster is expected to have an enhanced chemical stability. As a consequence, the substitution of an As atom would affect the chemical features of pure Ge_{n + 1} clusters.

The size dependence on the vertical ionization potential (VIP) for the most favorable geometries of Ge_{n + 1} and AsGe_n (n = 1–20) clusters are displayed in Fig. 7. For AsGe_n clusters, the VIP reveals an oscillating trend up to n = 14. All values are in the 6.2–8.8 eV range and decreases slowly as the cluster size increases and it is well known that when the VIP becomes smaller, the cluster will be more close to a metallic system. This means that the clusters of AsGe_n with size more than 6 atoms exhibit high metallic character which, consequently, these clusters can more easily lose one electron comparatively to the clusters of smaller size. The smallest VIP values are observed for AsGe₅, AsGe₆, AsGe₈, AsGe₉, AsGe₁₁, AsGe₁₃, AsGe₁₈ and AsGe₂₀ indicating that these clusters are more readily ionized than the others. The cluster AsGe₄ has large VIP value (8.809). In Fig. 8, we plotted the cluster size-dependent VEA for Ge_{n + 1} and AsGe_n clusters. It can be seen that the electron affinity reveal also an oscillating trend with an increasing behavior with the size, which means the larger clusters are expected to capture more easily electrons more easily. This means that the small AsGe_n clusters will become gradually unstable after they acquired an electron. The calculated values of VEA for the most stable AsGe_n clusters are much lower than the VIP values which indicating that these clusters can easily accept one electron.

Fig. 9 shows the evolution of the chemical hardness for the lowest energy structures of Ge_{n + 1} and AsGe_n (n = 1–20) clusters as a function of cluster size. Our calculations reveal that AsGe₄ clusters have the largest chemical hardness of 8.454 eV, confirming the better stability of this cluster as compared to the neighboring ones. Other local peaks are also observed for n = 12 and 17, leading to the conclusion that AsGe₁₂ and AsGe₁₇ will be less reactive than other cluster sizes. These clusters are very inert and can be considered as good candidates to the fabrication assembled cluster materials for application in nano-electronics and nanotechnologies. It has been established that chemical hardness is an electronic parameter that may characterize the relative stability of small clusters through the principle of maximum hardness (PMH) proposed by Pearson^{[39, 41]}. The clusters with high values of hardness are less reactive and more stable.

We have systematically investigated the structural, energetic and electronic properties of Ge_{n + 1} and AsGe_n (n = 1–20) clusters by means of DFT-based first principles quantum computations. The AsGe_n clusters adopted somehow similar structures as those obtained for Ge_{n + 1} except for n = 8, 10, 11, and 16, which significantly differed from their corresponding Ge_{n + 1}. In all cases, the As-doping atom was found to always be located on the surface. Their relative stabilities have been examined through the calculated binding energies, fragmentation energies, and second-order difference of energies. Their electronic features such as HOMO–LUMO energy gaps, vertical ionization potentials, vertical electron affinities, and chemical hardness were also examined.

Our theoretical study could give detailed and relevant information to deeply understand the possible effects of doping one single As atom on the properties of Ge_{n + 1} clusters. We believe this work will provide guidelines for future experimental work.

The authors thank Professor Ari Paavo Seitsonen (Ecole Normale Supérieure, ENS, Department of Chemistry, Paris, France) and Professor Bahayou Mohamed El Amine (Applied Mathematics Laboratory, LMA, Ouargla, Algeria) for all their advice and guidance.