Abstract
1. Introduction
Studying clusters of various chemical elements has become a modern research topic in both physical and chemical communities over the last four decades[
Numerous theoretical and experimental studies on Ge clusters have been published over the last decade[
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 Gen + 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 Gen + 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 AsGen clusters.
2. Computational methods
The electronic structure calculations of AsGenq (n = 1–20, q = 0, ± 1) clusters were performed using the density functional theory (DFT)[
We first searched for the lowest-energy structures of pure Gen + 1 clusters in the 1–20 atoms range by exploring various possibilities of isomers. Secondly, the most stable ground state structures obtained for Gen + 1 clusters were doped through substitution with one As atom. Then, the obtained AsGen clusters were optimized until reaching their ground states. In order to get lowest-energy structures of the AsGen 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 Gen + 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.
3. Results and discussion
3.1. Structural analysis
We report in Fig. 1 the lowest-energy structures obtained for Gen + 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[
Figure 1.(Color online) Most favorable structures together with their corresponding isomers for Ge
The most favorable geometries of AsGen (n = 1–20) clusters and their corresponding isomers are summarized in Fig. 2, whereas their energetic ordering is reported in Table 3. The AsGen clusters adopt somehow similar structures to their corresponding Gen + 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 C2v 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 C2v 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 C2v 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 C4v 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 C2v 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 C3v 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 Cs symmetry of the ground state isomer. The lowest-energy structure of AsGe9 cluster has Cs 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 Gen + 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.
Figure 2.(Color online) Most favorable structures and their corresponding isomers of AsGe
3.2. Relative stability
3.2.1. Binding energy
The size dependence on the binding energies per atom for the lowest energy structures of Gen + 1 and AsGen (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 AsGen, we observe that the binding energies are lower than those for Gen + 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 AsGen 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 AsGen clusters is larger than those of corresponding pure Gen + 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.
Figure 3.(Color online) Evolution of the binding energy per atom for the lowest energy structures of Ge
3.2.2. Fragmentation energy
Fig. 4 shows the plot of the size dependence on the fragmentation energies of Gen + 1 and AsGen (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.
Figure 4.(Color online) Evolution of the fragmentation energy of Ge
3.2.3. Second-order difference
The evolution of the second-order difference of energies for the most favorable structures of Gen + 1 and AsGen (n = 1–20) clusters as a function of the cluster size is plotted in Fig. 5. The curve shows pronounced peaks for AsGen 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 Gen + 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 AsGen structures with n = 3, 5, 11, 13 atoms correlates with the stability of the corresponding Gen + 1 structures, where the AsGen structure was maintained the same upon the incorporation of As dopant.
Figure 5.(Color online) Evolution of the second-order difference of energy for the lowest energy structures of Ge
3.3. Electronic properties
3.3.1. HOMO–LUMO gap
In order to obtain insight into the kinetic stability of AsGen clusters, we calculated and analyzed the HOMO–LUMO gap. In general, the reactivity of the cluster decreases with increasing the HOMO–LUMO gap[
Figure 6.(Color online) Evolution of the HOMO–LUMO gap for the lowest energy structures of Ge
3.3.2. Vertical ionization potential (VIP) and vertical electronic affinity (VEA)
The size dependence on the vertical ionization potential (VIP) for the most favorable geometries of Gen + 1 and AsGen (n = 1–20) clusters are displayed in Fig. 7. For AsGen 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 AsGen 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 Gen + 1 and AsGen 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 AsGen clusters will become gradually unstable after they acquired an electron. The calculated values of VEA for the most stable AsGen clusters are much lower than the VIP values which indicating that these clusters can easily accept one electron.
Figure 7.(Color online) Evolution of the vertical ionization potential (VIP) for the lowest energy structures of Ge
Figure 8.(Color online) Evolution of the vertical electron affinity (VEA) for the lowest energy structures of Ge
3.3.3. Chemical hardness
Fig. 9 shows the evolution of the chemical hardness for the lowest energy structures of Gen + 1 and AsGen (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[
Figure 9.(Color online) Evolution of the chemical hardness
4. Conclusion
We have systematically investigated the structural, energetic and electronic properties of Gen + 1 and AsGen (n = 1–20) clusters by means of DFT-based first principles quantum computations. The AsGen clusters adopted somehow similar structures as those obtained for Gen + 1 except for n = 8, 10, 11, and 16, which significantly differed from their corresponding Gen + 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 Gen + 1 clusters. We believe this work will provide guidelines for future experimental work.
Acknowledgments
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.
References
[1] J Wang, J G Han. The growth behaviors of the Zn-doped different sized germanium clusters: a density functional investigation. Chem Phys, 342, 253(2007).
[2] S Mahtout, Y Tariket. Electronic and magnetic properties of CrGe
[3] R W Schmude, K A Gingerich. Thermodynamic study of small silicon carbide clusters with a mass spectrometer. J Phys Chem A, 101, 2610(1997).
[4] P N Samanta, K K Das. Electronic structure, bonding, and properties of Sn
[5] J Jr Kingcade, K Gingerich. Knudsen effusion mass spectrometric investigation of palladium-germanium clusters. Inorg Chem, 28, 89(1989).
[6] P S Yadav, R K Yadav. Ab initio study of the physical properties of binary Si
[7] D Bandyopadhyay, M Kumar. The electronic structures and properties of transition metal-doped silicon nanoclusters: a density functional investigation. Chem Phys, 353, 170(2008).
[8] J G Han, F Hagelberg. Recent progress in the computational study of silicon and germanium clusters with transition metal impurities. J Comput Theor Nanosci, 6, 257(2009).
[9] S Bals, S Van Aert, C P Romero et al. Atomic scale dynamics of ultrasmall germanium clusters. Nat Commun, 3, 897(2012).
[10] C Siouani, S Mahtout, S Safer et al. Structure, stability, and electronic and magnetic properties of VGe
[11] M Brack. The physics of simple metal clusters: self-consistent jellium model and semiclassical approaches. Rev Mod Phys, 65, 677(1993).
[12] J G Han, P F Zhang, Q X Lic et al. A theoretical investigation of Ge
[13] A K Singh, V Kumar. Thorium encapsulated caged clusters of germanium: The Ge
[14] J Wang, J G Han. A computational investigation of copper-doped germanium and germanium clusters by the density-functional theory. J Chem Phys, 123, 244303(2005).
[15] W J Zhao, ng Wa, X Y. Geometries, stabilities, and magnetic properties of MnGe
[16] S Jaiswal, V Kumar. Growth behavior and electronic structure of neutral and anion ZrGe
[17] S Mahtout, C Siouani, F Rabilloud. Growth behavior and electronic structure of noble metal-doped germanium clusters. J Phys Chem A, 122, 662(2018).
[18] S Djaadi, K E Aiadi, S Mahtout. First principles study of structural, electronic and magnetic properties of SnGe
[19] P Ordejón, E Artacho, J M Soler. Self-consistent order-N density-functional calculations for very large systems. Phys Rev B, 53, R10441(1996).
[20] J M Soler, E Artacho, J D Gale et al. The siesta method for ab initio order-n materials simulation. J Phys Cond Matter, 14, 2745(2002).
[21] N Troullier, J L Martins. Efficient pseudopotentials for plane-wave calculations. Phys Rev B, 43, 1993(1991).
[22] J P Perdew, A Zunger. Self-interaction correction to density-functional approximations for many-electron systems. Phys Rev B, 23, 5048(1981).
[23] J P Perdew, K Burke, M Ernzerhof. Generalized gradient approximation made simple. Phys Rev Lett, 77, 3865(1996).
[24] D Bandyopadhyay, P Sen. Density functional investigation of structure and stability of Ge
[25] S Shi, Y Liu, C Zhang et al. A computational investigation of aluminum-doped germanium clusters by density functional theory study. Comput Theor Chem, 1054, 8(2015).
[26] N Kapila, I Garg, V K Jindal et al. First principle investigation into structural growth and magnetic properties in Ge
[27] J Wang, G Wang, J Zhao. Structure and electronic properties of Ge
[28] M Yoshida, J I Aihara. Validity of the weighted HOMO–LUMO energy separation as an index of kinetic stability for fullerenes with up to 120 carbon atoms. Phys Chem Chem Phys, 1, 227(1999).
[29] R G Parr, R G Pearson. Absolute hardness: companion parameter to absolute electronegativity. J Am Chem Soc, 105, 7512(1983).
[30] E Sosa-Hernández, P Alvarado-Leyva. Magnetic properties of stable structures of small binary Fe
[31] X Li, K Su, X Yang et al. Size-selective effects in the geometry and electronic property of bimetallic Au-Ge nanoclusters. Comput Theor Chem, 1010, 32(2013).
[32] J E Kingcade, H M Nagarathna-Naik, I Shim et al. Electronic structure and bonding of the dimeric germanium molecule from all-electron ab initio calculations and equilibrium measurements. J Phys Chem, 90, 2830(1986).
[33] S Nagendran, S S Sen, H W Roesky et al. RGe(I)Ge(I)R Compound (R = PhC(NtBu)2) with a Ge−Ge single bond and a comparison with the gauche conformation of hydrazine. Organometallics, 27, 5459(2008).
[34] G V Gadiyak, Y. N Morokov, A G Mukhachev et al. Electron density functional method for molecular system calculations. J Struct Chem, 22, 670(1982).
[35] J Wang, J G Han. Geometries, stabilities, and vibrational properties of bimetallic Mo2-doped Ge
[36] A Kant, B H Strauss. Atomization energies of the polymers of germanium, Ge2 to Ge7. J Chem Phys, 45, 822(1966).
[37] I S Vasiliev, S Öğüt, J R Chelikowsky. Ab initio calculations for the polarizabilities of small semiconductor clusters. Phys Rev Lett, 78, 4805(1997).
[38] G R Burton, C Xu, C C Arnold et al. Photoelectron spectroscopy and zero electron kinetic energy spectroscopy of germanium cluster anions. J Chem Phys, 104, 2757(1996).
[39] S Safer, S Mahtout, K Rezouali et al. Properties of neutral and charged cobalt-doped arsenic CoAs
[40] L Guo. The structure and energetic of AlAs
[41]
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