First principles study of structural, electronic and magnetic properties of SnGe<i><sub>

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1.
Introduction

In recent years, the clusters of different materials have attracted a great interest and their study has become a hot topic of research in both physics and chemistry. Because of their very small size, the evolution of their physical and chemical properties became very important for many applications in different fields such as optoelectronic materials, catalyst, nanoelectronic and spintronic^{[1, 2]}. Semiconductor clusters formed by the elements of group 14 have been also extensively investigated experimentally and theoretically in view of their fundamental understanding and technological applications. Germanium and silicon are the most important semiconductor elements with widespread applications. Previous studies have shown that the structure of small germanium and silicon clusters are very similar, while the larger ones are fundamentally different^[3]. In this work, we studied germanium clusters. During the last four decades, pure and doped germanium clusters have been intensively studied experimentally^[4–10] and theoretically^[11–17]. Zhao and Wang^[18] studied geometries, stabilities and electronics properties of FeGe_n (n = 9–16) clusters. They found that the encapsulation of one iron atom Fe in the pure Ge_n clusters contributes in strengthening the stabilities of the germanium framework and the HOMO-LUMO gaps of the FeGe_n clusters which are usually smaller than those of the corresponding pure germanium clusters. In their study concerned with the properties of aluminum-doped small germanium clusters, Shi et al.^[19] showed that the lowest energy structures of the AlGe_n are similar to lowest energy structures of the pure corresponding germanium clusters. Li et al.^[20] reported the effect of one atom of gold Au in anionic germanium clusters AuGe_n^– (n = 1–13). They found that the clusters with n = 12 are the most stable ones compared to the other size. Electronic and magnetic properties of Mn-, Co- and Ni-doped small germanium clusters have been investigated using spin polarized density functional theory by Kapila et al.^[21]. They found that the Ni-doped Ge_n clusters are the more stable when compared to Co- and Mn-doped Ge_n clusters. Tang et al.^[22] studied the structure and stability of the 3d transition metal endohedral Ge₁₂M (M = Sc–Ni) clusters. They found that all Ge₁₂M clusters have the doping energies (DEs) and the energies gap (Egs) comparable to the isolated compounds, which indicates their remarkably kinetic stability and the possibility for isolation. Mahtout et al.^[2] calculated the electronic and magnetic properties of medium-sized CrGe_n (15 ≤ n ≤ 29) clusters by using first principles DFT investigation. They showed that the structures of Ge_n₊₁ and CrGe_n become more compact and switch to near-spherical structure with one or more core atoms and the clusters with size 17, 19, and 22 exhibit high stability. Katircioglu^[23] reported that the C–C bonds are privileged over Ge–C bonds and Ge–Ge binding for different values of n and m in Ge_nC_m–n clusters. Recently, small SnGe_n clusters have been studied in the range n < 5. Andzelm et al.^[24] performed a DFT study on GeSn monomer, and they reported on the spectroscopic constants and electronic structure of the diatomic molecules GeSn in their low-lying electronic states. They showed that the local spin-density model potentials (LSD) used in their study form a good description of chemical bonding and spectroscopic properties of molecules. Schmude and Gingenich^[25] presented an experimental study on the small germanium–tin clusters and they find the atomization enthalpies of the GeSn, GeSn₂, Ge₂Sn, Ge₃Sn, Ge₄Sn molecules. A DFT investigation of Sn_mGe_n (m + n ≤ 5) binary clusters has been performed by Samanta and Das^[1]. They showed that the larger mixed tin and germanium clusters have higher binding energies and the larger HOMO–LUMO energy gaps, which implies their high stability. Hanet al.^[26] presented theoretical investigation of very small SnGe_n (n = 1–4) clusters. They calculated equilibrium geometry, enthalpy of reaction and natural population and they found that natural populations of these clusters indicate a charge transfer from Sn to Ge atoms. In light of the previous studies, to the best of our knowledge, systematic and theoretical investigation on neutral and charged tin-doped germanium SnGe_n clusters with range of 5 to 17 atoms have not been reported so far. The main motivation behind the present work is to study the inclusion effect of one Sn on the electronic and magnetic properties of different isomers of Ge_n₊₁ clusters in the size range n = 1–17 atoms and their evolution as a function of the size by using the density functional theory calculations. The properties and the associated adiabatic energies of charged tin-doped germanium are also studied. Our investigation will provide noteworthy contribution for theoretical and experimental studies. The paper is organized as follows: we describe the computational details in Section 2, Section 3 is devoted to discuss the obtained results and in Section 4 we draw some conclusions.

2.
Method

In this study, ab initio molecular orbital and density functional theory (DFT) calculations are performed by using the SIESTA software package^[27–29] (the Spanish Initiative for Electronic Simulation with Thousands of Atoms). The density functional is treated by generalized gradient approximation (GGA) with exchange correlation potential developed by Perdew and Zanger^[30] and Perdew, Burke and Ernserhof (PBE)^[31]. Self-consistent field procedures are carried out with a convergence (SZ) basis set for Sn and Ge atoms. The ionic core criterion of 10^–4 a.u. on the total energy and electron density. A big cubic supercell with dimension of 40 ? was used to create sufficient vacuum space to eliminate the image interactions. The k = 0 (Γ) point approximation was used in Brillouin zone sampling. During simulation, volume of the system was kept constant and a single ? potential were Bylander form^[32]. The energy cut-off of 150 Ry was represented by norm-conserving Troullier-Martins^[33] and non local pseudopotentiels factorized in the Kleinman and PAO. EnergyShift is taken equal to 50 meV. Conjugate gradient method within Hellmann Feynman forces was used and all the forces are less than 10^–2 eV/?. In order to find the global minimum structures of SnGe_n clusters, at first we have optimized several isomers of pure germanium clusters with size of 2–18 atoms. Second, a great number of isomers for doped SnGe_n were considered. We have initially relaxed different possible isomers of neutral SnGe_n clusters. Then, the best obtained structures have been considered in their anionic and cationic configurations and relaxed. After conducting a simulation process, a comparative study between the properties of neutral and charged clusters was performed. In the discussion, we consider only the lowest energy isomers determined in our optimizations. In order to test the method used in geometry optimization, with respect to the exchange-correlation functional to the size and to the cutoff radii of basis sets, we performed calculations on Ge₂ and Ge₃ clusters. Our results shown in Table 1 are in good agreement with theoretical and experimental results of the literature.

Cluster	Our works				Theoretical values				Experimental values
Cluster	a	E_b	AEA	AIP	a	E_b	AEA	AIP	a	E_b	AEA	AIP
Ge²	2.625	1.28	2.060	7.753	2.548^d	1.3^e	2.10ⁱ	7.9^j	/	1.32^g	2.035^g	7.58–7.76^j
					2.54^b	1.31^f				1.41^m	2.074^h
					2.413^c	1.32^j
					2.41^a	1.230^l
					2.61^k	1.812^k
Ge³	2.476	1.86	1.907	8.126	2.428ⁱ	1.92^e	2.09ⁱ	7.92^j	/	2.24ⁿ	2.23^g	7.97–8.09^j
					2.400^l	1.98^d				2.15^m	2.23^h
					2.378^d	2^a
					2.296ⁱ	2.04^c
^a Ref. [34]. ^b Ref. [35]. ^c Ref. [19]. ^d Ref. [36]. ^e Ref. [37]. ^f Ref. [38]. ^g Ref. [4]. ^h Ref.[5]. ^I Ref.[11]. ^j Ref.[39]. ^k Ref. [2]. ^l Ref. [12]. ^m Ref. [40]. ⁿ Ref. [41].

Table1.
Average bond length a (?), binding energy E_b (eV), adiabatic electronic affinity (AEA) (eV) and adiabatic ionisation potential (AIP) (eV) for Ge₂ and Ge₃ clusters.

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Cluster	Our works				Theoretical values				Experimental values
Cluster	a	E_b	AEA	AIP	a	E_b	AEA	AIP	a	E_b	AEA	AIP
Ge²	2.625	1.28	2.060	7.753	2.548^d	1.3^e	2.10ⁱ	7.9^j	/	1.32^g	2.035^g	7.58–7.76^j
					2.54^b	1.31^f				1.41^m	2.074^h
					2.413^c	1.32^j
					2.41^a	1.230^l
					2.61^k	1.812^k
Ge³	2.476	1.86	1.907	8.126	2.428ⁱ	1.92^e	2.09ⁱ	7.92^j	/	2.24ⁿ	2.23^g	7.97–8.09^j
					2.400^l	1.98^d				2.15^m	2.23^h
					2.378^d	2^a
					2.296ⁱ	2.04^c
^a Ref. [34]. ^b Ref. [35]. ^c Ref. [19]. ^d Ref. [36]. ^e Ref. [37]. ^f Ref. [38]. ^g Ref. [4]. ^h Ref.[5]. ^I Ref.[11]. ^j Ref.[39]. ^k Ref. [2]. ^l Ref. [12]. ^m Ref. [40]. ⁿ Ref. [41].

3.
Results and discussions

3.1
Structural properties

In cluster physics, one of the most important things in studying the properties of clusters is to determine their ground states geometries. The most obtained stable structure and their corresponding isomers for pure and tin-doped germanium clusters are shown in Figs. 1 and 2, respectively. In these figures, (a) isomer is the most stable structure. The energetic ordering of isomers for Ge_n₊₁ and SnGe_n (n = 1–17) clusters are given in Tables 2 and 3, respectively. In these tables, the parameters of the most stable structures for each size are reported in bold character. In Table 4, we give the average Ge–Ge and Sn–Ge bond lengths of the lowest energy structures of SnGe_n^{(0, ±1)} clusters. Fig. 3 presents the most stable structures of anionic and cationic SnGe_n^(±1) clusters and their parameters are described in Table 5. In the case of clusters with two atoms, we obtained a SnGe monomer with bond length of 2.805 ?, which is in good agreement with the theoretical value^[33] of 2.753 ?. The binding energy equals 1.19 eV/atom, which is smaller than that of Ge₂ monomer (1.28 eV/atom). For charged monomer, the bond length is 2.910 and 2.495 ? for SnGe⁺ and SnGe^–, respectively, while the binding energy is 1.285 and 1.819 eV/atom, respectively. The cationic monomer is more stable than the neutral and anionic monomers. The triangular structure is the lowest energy structure for Ge₃ with symmetry C_2v and average bond length of Ge–Ge of 2.476 ?. For SnGe₂ cluster, which has a triangular structure, with C_s symmetry, the average bond lengths Ge–Ge and Sn–Ge are 2.472 and 2.658 ?, respectively. The geometric structure of the anion cluster of SnGe₂ is also triangular, whereas its cation cluster is angular with C_s symmetry. For Ge₄, the most stable structure is the rhombus (D2h) with the bond length of 2.714 ? and the binding energy E_b = 2.25 eV/atom. The same structure is obtained by Wang et al.^[12] in their theoretical study and Li et al.^[39] in their experimental study on the Ge_n systems. In the case of SnGe₃ cluster, the ground state structure showed a C_2v symmetry with Sn–Ge and Ge–Ge bond distance of 2.857 and 2.741 ?, respectively. Rhombus structure is also found to be the most stable structure for charged SnGe₃ cluster. The ground state of Ge₅ cluster is a trigonal bipyramid with D_3h symmetry and average bond length of 2.668 ?. Our calculations indicate that the SnGe₄ cluster has also trigonal bipyramid with C_2v structure. Their Ge–Ge bond length increases to 2.665 ? and Sn–Ge bond length is found to be 2.852 ?.

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Figure3.
(Color online) Ground state structures of cationic and anionic SnGe_n^(±1) (n = 1–17) clusters.

Size (n+1)	Symmetry group	E_b (eV/atm)	ΔE (eV)	a (?)
1	(a) D_∞h	1.28	0.367	2.625
2	(a) C_2v	1.86	1.368	2.476
3	(a) D_2h	2.25	1.152	2.714
4	(a) D_3h	2.44	1.815	2.668
5	(a) C_2v	2.55	1.650	2.823
5	(b) C_s	2.49	1.321	2.811
6	(a) D_5h	2.66	1.557	2.900
6	(b) C_2v	2.59	2.111	2.722
7	(a) C_s	2.64	1.179	2.796
	(b) C₂	2.61	1.143	2.837
	(c) C_2h	2.61	1.153	2.836
8	(a) C_2v	2.70	1.588	3.000
8	(b) C₁	2.64	1.306	2.854
9	(a) C_3v	2.80	1.609	2.857
	(b) C_s	2.73	1.415	2.854
	(c) C_2v	2.60	0.817	2.849
10	(a) C_s	2.76	1.148	2.857
	(b) C₁	2.76	1.140	2.857
	(c) C₂	2.73	1.143	2.944
11	(a) C_2v	2.76	1.536	2.884
	(b) C₁	2.75	1.113	2.940
	(c) C_s	2.71	1.299	2.855
12	(a) C₁	2.77	0.982	2.943
	(b) C₂	2.75	1.235	2.836
	(c) C_s	2.75	1.096	2.788
13	(a) C_s	2.83	1.372	2.867
	(b) C₁	2.80	1.335	2.888
	(c) C₂	2.76	1.014	2.871
14	(a) C_2v	2.82	0.864	2.925
	(b) C₁	2.78	1.116	2.768
	(c) C_s	2.77	1.163	2.862
15	(a) C_2h	2.83	1.164	2.893
	(b) C₁	2.78	0.978	2.852
	(c) C₁	2.77	1.172	2.825
	(d) C₁	2.77	1.138	2.875
16	(a) C_s	2.82	0.884	2.893
	(b) C₁	2.82	0.908	2.861
	(c) C₁	2.82	1.116	2.836
17	(a) C₁	2.81	1.322	2.822
17	(b) C₁	2.79	0.626	2.845

Table2.
Symmetry group, binding energy per atom E_b, HOMO-LUMO gap ΔE and average bond length for pure Ge_n+1 (n = 1–17) clusters.

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Size (n+1)	Symmetry group	E_b (eV/atm)	ΔE (eV)	a (?)
1	(a) D_∞h	1.28	0.367	2.625
2	(a) C_2v	1.86	1.368	2.476
3	(a) D_2h	2.25	1.152	2.714
4	(a) D_3h	2.44	1.815	2.668
5	(a) C_2v	2.55	1.650	2.823
5	(b) C_s	2.49	1.321	2.811
6	(a) D_5h	2.66	1.557	2.900
6	(b) C_2v	2.59	2.111	2.722
7	(a) C_s	2.64	1.179	2.796
	(b) C₂	2.61	1.143	2.837
	(c) C_2h	2.61	1.153	2.836
8	(a) C_2v	2.70	1.588	3.000
8	(b) C₁	2.64	1.306	2.854
9	(a) C_3v	2.80	1.609	2.857
	(b) C_s	2.73	1.415	2.854
	(c) C_2v	2.60	0.817	2.849
10	(a) C_s	2.76	1.148	2.857
	(b) C₁	2.76	1.140	2.857
	(c) C₂	2.73	1.143	2.944
11	(a) C_2v	2.76	1.536	2.884
	(b) C₁	2.75	1.113	2.940
	(c) C_s	2.71	1.299	2.855
12	(a) C₁	2.77	0.982	2.943
	(b) C₂	2.75	1.235	2.836
	(c) C_s	2.75	1.096	2.788
13	(a) C_s	2.83	1.372	2.867
	(b) C₁	2.80	1.335	2.888
	(c) C₂	2.76	1.014	2.871
14	(a) C_2v	2.82	0.864	2.925
	(b) C₁	2.78	1.116	2.768
	(c) C_s	2.77	1.163	2.862
15	(a) C_2h	2.83	1.164	2.893
	(b) C₁	2.78	0.978	2.852
	(c) C₁	2.77	1.172	2.825
	(d) C₁	2.77	1.138	2.875
16	(a) C_s	2.82	0.884	2.893
	(b) C₁	2.82	0.908	2.861
	(c) C₁	2.82	1.116	2.836
17	(a) C₁	2.81	1.322	2.822
17	(b) C₁	2.79	0.626	2.845

Size (n)	Symmetry group	E_b (eV/atom)	ΔE (eV)	VEA (eV)	VIP (eV)	η (eV)	μ (μ_b)
1	(a) C_∞v	1.19	0.344	1.199	7.383	6.184	2.000
2	(a) C_s	1.81	1.311	1.441	7.982	6.541	0.000
	(b) C_∞v	1.71	1.643	1.264	8.719	7.455	0.000
	(c) C_2v	1.80	0.984	1.702	7.936	6.234	0.000
3	(a) C_2v	2.20	0.227	1.715	8.039	6.324	0.000
3	(b) C_2v	1.79	1.132	2.386	7.072	4.686	0.000
4	(a) C_2v	2.40	1.739	1.649	8.912	7.263	0.000
	(b) C_3v	2.35	1.556	1.255	8.447	7.192	0.000
	(c) C_s	2.19	0.499	2.687	7.367	4.680	2.000
	(d) C_2v	2.14	0.889	2.959	6.840	3.881	2.000
5	(a) C_s	2.53	1.663	1.845	8.022	6.177	0.000
	(b) C_s	2.48	1.450	2.056	7.798	8.742	0.000
	(c) C_s	2.44	1.228	1.873	7.786	5.913	0.000
6	(a) C₁	2.63	1.853	2.064	8.241	6.177	0.000
	(b) C₁	2.61	1.512	2.357	8.153	5.796	0.000
	(c) C_2v	2.53	1.451	1.590	7.856	6.266	0.000
7	(a) C₁	2.62	1.197	2.059	7.538	5.479	0.000
	(b) C₁	2.59	1.149	2.076	7.416	5.340	0.000
	(c) C_s	2.58	0.947	2.047	7.179	5.132	0.000
8	(a) C₁	2.68	1.486	1.749	7.745	5.996	0.000
8	(b) C₁	2.68	1.515	1.978	7.425	5.447	0.000
9	(a) C_s	2.77	1.551	2.306	7.714	5.408	0.000
9	(b) C₁	2.69	1.434	2.197	7.497	5.300	0.000
10	(a) C_s	2.73	1.202	2.291	7.204	4.913	0.000
	(b) C₁	2.72	1.031	2.378	7.133	4.755	0.000
	(c) C₁	2.71	1.121	2.511	7.324	4.813	0.000
11	(a) C₁	2.74	1.529	2.225	7.321	5.096	0.000
	(b) C_5v	2.70	1.155	2.583	7.388	4.805	0.000
	(c) C_s	2.67	1.104	2.667	7.359	4.692	0.000
	(d) C₁	2.64	1.027	3.628	6.242	2.614	0.000
12	(a) C₁	2.73	0.992	2.230	7.207	4.977	0.000
	(b) C_2v	2.70	1.014	2.822	7.421	4.599	0.000
	(c) D_3d	2.62	1.046	2.754	6.804	4.050	0.000
13	(a) C₁	2.81	1.353	2.556	7.310	4.754	0.000
	(b) C₁	2.78	1.310	2.585	7.329	4.744	0.000
	(c) C₁	2.60	0.554	2.913	6.951	4.038	0.000
	(d) C_6v	2.72	0.517	3.115	7.122	4.007	0.000
14	(a) C_s	2.79	0.950	2.889	6.396	3.507	0.000
	(b) C₁	2.75	1.078	2.721	7.125	4.404	0.000
	(c) D_6h	2.58	0.010	2.916	6.358	3.442	0.002
	(d) C₁	2.73	1.097	2.690	7.070	4.380	0.000
15	(a) C₂	2.80	1.031	3.022	7.357	4.335	0.000
	(b) C₁	2.77	0.997	2.758	7.026	4.268	0.000
	(c) C_s	2.69	0.591	2.975	6.816	3.841	0.000
	(d) C_2v	2.64	0.520	2.758	6.424	3.666	0.000
16	(a) C₁	2.81	0.899	2.780	6.939	4.159	0.000
	(b) C₁	2.80	0.994	2.953	7.069	4.116	0.000
	(c) C_2h	2.74	0.537	2.571	6.348	3.777	0.000
17	(a) C₁	2.80	1.332	2.664	7.136	4.472	0.000
	(b) C_s	2.71	1.018	2.747	6.863	4.116	0.000
	(c) C_2v	2.69	1.015	2.219	6.826	4.607	0.000

Table3.
Symmetry group, binding energy per atom E_b, HOMO-LUMO gap ΔE, vertical electronic affinity (VEA), vertical ionisation potential (VIP), chemical hardness η and total spin magnetic moments μ for SnGen (n = 1–17) clusters.

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Size (n)	Symmetry group	E_b (eV/atom)	ΔE (eV)	VEA (eV)	VIP (eV)	η (eV)	μ (μ_b)
1	(a) C_∞v	1.19	0.344	1.199	7.383	6.184	2.000
2	(a) C_s	1.81	1.311	1.441	7.982	6.541	0.000
	(b) C_∞v	1.71	1.643	1.264	8.719	7.455	0.000
	(c) C_2v	1.80	0.984	1.702	7.936	6.234	0.000
3	(a) C_2v	2.20	0.227	1.715	8.039	6.324	0.000
3	(b) C_2v	1.79	1.132	2.386	7.072	4.686	0.000
4	(a) C_2v	2.40	1.739	1.649	8.912	7.263	0.000
	(b) C_3v	2.35	1.556	1.255	8.447	7.192	0.000
	(c) C_s	2.19	0.499	2.687	7.367	4.680	2.000
	(d) C_2v	2.14	0.889	2.959	6.840	3.881	2.000
5	(a) C_s	2.53	1.663	1.845	8.022	6.177	0.000
	(b) C_s	2.48	1.450	2.056	7.798	8.742	0.000
	(c) C_s	2.44	1.228	1.873	7.786	5.913	0.000
6	(a) C₁	2.63	1.853	2.064	8.241	6.177	0.000
	(b) C₁	2.61	1.512	2.357	8.153	5.796	0.000
	(c) C_2v	2.53	1.451	1.590	7.856	6.266	0.000
7	(a) C₁	2.62	1.197	2.059	7.538	5.479	0.000
	(b) C₁	2.59	1.149	2.076	7.416	5.340	0.000
	(c) C_s	2.58	0.947	2.047	7.179	5.132	0.000
8	(a) C₁	2.68	1.486	1.749	7.745	5.996	0.000
8	(b) C₁	2.68	1.515	1.978	7.425	5.447	0.000
9	(a) C_s	2.77	1.551	2.306	7.714	5.408	0.000
9	(b) C₁	2.69	1.434	2.197	7.497	5.300	0.000
10	(a) C_s	2.73	1.202	2.291	7.204	4.913	0.000
	(b) C₁	2.72	1.031	2.378	7.133	4.755	0.000
	(c) C₁	2.71	1.121	2.511	7.324	4.813	0.000
11	(a) C₁	2.74	1.529	2.225	7.321	5.096	0.000
	(b) C_5v	2.70	1.155	2.583	7.388	4.805	0.000
	(c) C_s	2.67	1.104	2.667	7.359	4.692	0.000
	(d) C₁	2.64	1.027	3.628	6.242	2.614	0.000
12	(a) C₁	2.73	0.992	2.230	7.207	4.977	0.000
	(b) C_2v	2.70	1.014	2.822	7.421	4.599	0.000
	(c) D_3d	2.62	1.046	2.754	6.804	4.050	0.000
13	(a) C₁	2.81	1.353	2.556	7.310	4.754	0.000
	(b) C₁	2.78	1.310	2.585	7.329	4.744	0.000
	(c) C₁	2.60	0.554	2.913	6.951	4.038	0.000
	(d) C_6v	2.72	0.517	3.115	7.122	4.007	0.000
14	(a) C_s	2.79	0.950	2.889	6.396	3.507	0.000
	(b) C₁	2.75	1.078	2.721	7.125	4.404	0.000
	(c) D_6h	2.58	0.010	2.916	6.358	3.442	0.002
	(d) C₁	2.73	1.097	2.690	7.070	4.380	0.000
15	(a) C₂	2.80	1.031	3.022	7.357	4.335	0.000
	(b) C₁	2.77	0.997	2.758	7.026	4.268	0.000
	(c) C_s	2.69	0.591	2.975	6.816	3.841	0.000
	(d) C_2v	2.64	0.520	2.758	6.424	3.666	0.000
16	(a) C₁	2.81	0.899	2.780	6.939	4.159	0.000
	(b) C₁	2.80	0.994	2.953	7.069	4.116	0.000
	(c) C_2h	2.74	0.537	2.571	6.348	3.777	0.000
17	(a) C₁	2.80	1.332	2.664	7.136	4.472	0.000
	(b) C_s	2.71	1.018	2.747	6.863	4.116	0.000
	(c) C_2v	2.69	1.015	2.219	6.826	4.607	0.000

Cluster size (n)	SnGe_n		SnGe_n⁺		SnGe_n^–
Cluster size (n)	a_Ge-Ge	a_Sn-Ge	a_Ge-Ge	a_Sn-Ge	a_Ge-Ge	a_Sn-Ge
1	/	2.805	/	2.910	/	2.495
2	2.472	2.658	2.626	2.833	2.551	2.900
3	2.741	2.857	2.656	2.884	2.689	2.844
4	2.665	2.852	2.691	3.026	2.683	2.861
5	2.850	2.911	2.867	2.974	2.856	3.034
6	2.714	2.927	2.912	3.010	2.452	3.047
7	2.581	3.130	2.892	3.154	2.862	3.042
8	2.939	3.062	2.978	3.107	2.908	2.999
9	2.851	3.059	2.498	3.008	3.022	2.976
10	2.846	3.088	3.033	3.071	2.848	3.002
11	2.866	3.052	2.873	3.086	2.641	2.996
12	2.904	3.059	2.931	3.078	2.901	3.029
13	2.860	3.083	2.787	3.049	2.850	3.057
14	2.980	3.038	2.855	3.054	2.917	2.969
15	2.906	2.907	2.925	2.929	2.882	2.931
16	2.895	3.037	3.081	3.029	2.897	3.021
17	2.865	2.996	2.837	2.986	2.854	3.002

Table4.
Average bond length a_Ge-Ge and a_Sn-Ge for neutral, cationic and anionic SnGe_n^{(0 ± 1)} (n = 1–17) clusters.

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Cluster size (n)	SnGe_n		SnGe_n⁺		SnGe_n^–
Cluster size (n)	a_Ge-Ge	a_Sn-Ge	a_Ge-Ge	a_Sn-Ge	a_Ge-Ge	a_Sn-Ge
1	/	2.805	/	2.910	/	2.495
2	2.472	2.658	2.626	2.833	2.551	2.900
3	2.741	2.857	2.656	2.884	2.689	2.844
4	2.665	2.852	2.691	3.026	2.683	2.861
5	2.850	2.911	2.867	2.974	2.856	3.034
6	2.714	2.927	2.912	3.010	2.452	3.047
7	2.581	3.130	2.892	3.154	2.862	3.042
8	2.939	3.062	2.978	3.107	2.908	2.999
9	2.851	3.059	2.498	3.008	3.022	2.976
10	2.846	3.088	3.033	3.071	2.848	3.002
11	2.866	3.052	2.873	3.086	2.641	2.996
12	2.904	3.059	2.931	3.078	2.901	3.029
13	2.860	3.083	2.787	3.049	2.850	3.057
14	2.980	3.038	2.855	3.054	2.917	2.969
15	2.906	2.907	2.925	2.929	2.882	2.931
16	2.895	3.037	3.081	3.029	2.897	3.021
17	2.865	2.996	2.837	2.986	2.854	3.002

(A) SnGe_n⁺						(B) SnGe_n^-
Cluster size (n)	Symmetry group	E_b (eV/atm)	ΔE (eV)	μ(μ_b)	AIP (eV)	Symmetry group	E_b (eV/atm)	ΔE (eV)	μ (μ_b)	AEA (eV)
1	C_∞v	1.285	1.476	3.000	7.360	C_∞v	1.819	0.247	1.000	1.451
2	C_s	1.761	0.410	1.000	7.706	C_s	2.365	0.601	1.000	1.850
3	C_2v	2.156	0.674	1.000	7.729	C_2v	2.640	0.990	1.000	1.948
4	C_2v	2.316	0.405	1.000	7.971	C_2v	2.833	0.930	1.000	2.357
5	C_2v	2.515	1.027	1.000	7.624	C_2v	2.878	0.892	1.000	2.300
6	C_2v	2.597	0.530	1.000	7.796	C_2v	2.956	1.284	0.996	2.460
7	C₁	2.655	0.894	0.999	7.252	C₁	2.909	0.827	1.000	2.525
8	C₁	2.711	0.420	0.998	7.277	C₁	2.975	0.430	0.998	2.844
9	C_s	2.760	0.372	0.997	7.597	C_s	3.007	0.669	0.999	2.611
10	C₁	2.775	0.944	0.997	7.012	C_s	2.945	0.376	0.999	2.596
11	C₁	2.772	0.430	0.992	7.158	C₁	2.934	0.376	0.994	2.521
12	C₁	2.771	0.435	0.995	7.086	C₁	2.958	0.061	0.997	3.100
13	C₁	2.836	0.535	0.992	7.139	C₁	2.985	0.174	0.976	2.682
14	C_s	2.828	0.650	0.999	6.939	C_s	2.992	0.426	0.985	3.244
15	C₂	2.829	0.635	0.994	7.046	C₂	2.991	0.446	0.945	3.278
16	C₁	2.854	0.343	0.991	6.824	C₁	2.978	0.151	0.966	3.022
17	C₁	2.834	0.194	0.969	6.960	C₁	2.963	0.122	0.976	3.110

Table5.
(A) Symmetry group, binding energy per atom E_b (eV/atom), HOMO-LUMO gap ΔE (eV), total spin magnetic moments μ(μ_B), adiabatic ionisation potential (AIP) (eV) for cationic SnGe_n⁺ (n = 1?17) clusters. (B) Symmetry group, binding energy per atom E_b, HOMO-LUMO gap ΔE, total spin magnetic moments μ, adiabatic electron affinity (AEA) for anionic SnGe_n^– (n = 1?17) clusters.

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(A) SnGe_n⁺						(B) SnGe_n^-
Cluster size (n)	Symmetry group	E_b (eV/atm)	ΔE (eV)	μ(μ_b)	AIP (eV)	Symmetry group	E_b (eV/atm)	ΔE (eV)	μ (μ_b)	AEA (eV)
1	C_∞v	1.285	1.476	3.000	7.360	C_∞v	1.819	0.247	1.000	1.451
2	C_s	1.761	0.410	1.000	7.706	C_s	2.365	0.601	1.000	1.850
3	C_2v	2.156	0.674	1.000	7.729	C_2v	2.640	0.990	1.000	1.948
4	C_2v	2.316	0.405	1.000	7.971	C_2v	2.833	0.930	1.000	2.357
5	C_2v	2.515	1.027	1.000	7.624	C_2v	2.878	0.892	1.000	2.300
6	C_2v	2.597	0.530	1.000	7.796	C_2v	2.956	1.284	0.996	2.460
7	C₁	2.655	0.894	0.999	7.252	C₁	2.909	0.827	1.000	2.525
8	C₁	2.711	0.420	0.998	7.277	C₁	2.975	0.430	0.998	2.844
9	C_s	2.760	0.372	0.997	7.597	C_s	3.007	0.669	0.999	2.611
10	C₁	2.775	0.944	0.997	7.012	C_s	2.945	0.376	0.999	2.596
11	C₁	2.772	0.430	0.992	7.158	C₁	2.934	0.376	0.994	2.521
12	C₁	2.771	0.435	0.995	7.086	C₁	2.958	0.061	0.997	3.100
13	C₁	2.836	0.535	0.992	7.139	C₁	2.985	0.174	0.976	2.682
14	C_s	2.828	0.650	0.999	6.939	C_s	2.992	0.426	0.985	3.244
15	C₂	2.829	0.635	0.994	7.046	C₂	2.991	0.446	0.945	3.278
16	C₁	2.854	0.343	0.991	6.824	C₁	2.978	0.151	0.966	3.022
17	C₁	2.834	0.194	0.969	6.960	C₁	2.963	0.122	0.976	3.110

This structure is in agreement with Samanta et al.^[1] results. For the charged cluster, the capped tetrahedron structure is the most stable structure with C_2v symmetry. The energy binding of SnGe₄⁺ is 2.316 eV/atom, which is smaller than that of SnGe₄^– (2.833 eV/atom). The capped trigonal bipyramid with (C_2v) symmetry is obtained for Ge₆ cluster, in agreement with the result of Li et al.^[39] For n = 5, SnGe₅ (a) is the most stable isomer with C_s symmetry. Their corresponding Ge–Ge and Sn–Ge bonds length are 2.850 and 2.911 ?, respectively. The same form is obtained by Wang and Han^[42]. The capped trigonal bipyramid configuration with C_2v is also stable for SnGe₅⁺. However, the anion SnGe₅^– is a distorted octahedron with (C_2v) symmetry. For Ge₇, the pentagonal bipyramid with D_5h symmetry and the lowest binding energy of 2.66 eV/atom is suggested. Our results are in good agreement with previous theoretical studies^{[3, 12, 39]}. Three different isomers are found for SnGe₆. The most stable one is SnGe₆ (a) with C_s symmetry, bond lengths Ge–Ge and Sn–Ge of 2.714 and 2.927 ?, respectively. The pentagonal bipyramid is the most stable structure for SnGe₆^– and SnGe₆⁺. For n = 8, the face-capped pentagonal bipyramid is obtained for pure Ge₈ cluster with C_s symmetry, binding energy of 2.64 eV/atom and average bond length of 2.796 ?. The same structure is reported by Wang et al.^[12] A similar structure as Ge₈ is obtained in the case of SnGe₇ and SnGe₇± with C₁ symmetry. The Ge–Ge bond length is much larger in the case of SnGe₇⁺ and SnGe₇^– (2.892 and 2.862 ?) clusters compared to the corresponding neutral SnGe₇ cluster (2.581 ?). For Ge₉, the bicapped pentagonal bipyramid Ge₉ (a) with C_2v symmetry is the most stable structure with binding energy of 2.70 eV/atom and average bond length Ge–Ge of 3?. In the corresponding doped cluster, the most stable structure is SnGe₈ (a) with C_s symmetry. Its Ge–Ge and Sn–Ge average distances are 2.939 and 3.062 ?, respectively. The same structure and symmetry is found for cationic SnGe₈⁺. However, the anionic SnGe₈^– cluster presents a different structure with C₁ symmetry. The lowest energy structure, among the three best structures observed in the case of Ge₁₀ cluster, is Ge₁₀ (a) with C_3v symmetry, which is in good agreement with Wang et al.^[12] results. In the doped clusters case, the ground state structure is SnGe₉ (a) with C_s symmetry. The Sn atom occupies a peripheral position. The average Ge–Ge and Sn–Ge distances are 2.851 and 3.059 ?, respectively. The results indicate that the SnGe₉⁺ and SnGe₉^– structure are completely different to the neutral SnGe₉ structure. The average Ge–Ge distance of cationic cluster 0.353 ? is smaller than that in the neutral SnGe₉. For Ge₁₁, the most stable structure is Ge₁₁ (a) with C_s symmetry. Substituting Ge by Sn leads to a somewhat similar structure. The ground state structure of SnGe₁₀ (a) is given in Fig. 2 with C_s symmetry. Its calculated binding energy is 2.73 eV/atom, while the bond lengths Ge–Ge and Sn–Ge are 2.846 and 3.088 ?, respectively. We observe that the obtained results suggest that the majority lowest energy clusters of pure germanium are layered structure beginning n = 12. In Fig. 1, we show that the ground state structure of Ge₁₂ has stacked structure with C_2v symmetry. Substituting Ge by Sn does not perturb significantly the structure because the structure of SnGe₁₁ is very similar to that of Ge₁₂. The binding energy of SnGe₁₁ is only 0.02 eV/atom, which is less than that of Ge₁₂. The stacked structure is also stable for SnGe₁₁⁺ species with C₁ symmetry. The layered structure has been found also for Ge₁₃. It consists in a triangle Ge₃ and bicapped square antiprism Ge₁₀, in 1-5-4-3 layers. Similar layered structure Ge₁₄ (a) is obtained for Ge₁₄ with 1-5-4-4 layered. In this case, the Ge₃ unit of the best isomer Ge₁₃ is replaced by a rhombus Ge₄. The lowest energy structures of Ge₁₅, Ge₁₆, and Ge₁₇ are stacked structures with 1-5-3-5-1, 1-5-4-5-1 and 1-5-5-5-1 layers, respectively. These layered structures have also been found by Wang et al.^[12]. Two isomers with spherical configuration and one Ge core atom are found for Ge₁₆ and Ge₁₈, as shown in Fig. 1. The most stable structure Ge₁₈ (a) of Ge₁₈ consists in three connected pentagonal parties Ge₇, dimer Ge₂ and capped tetragonal prism Ge₉ but this isomer is not a layered structure. In all the SnGe_n isomers clusters, with n < 12 size, the Sn atom occupies a peripheral position. In n > 12 case, the Sn atom can occupy a core position. For example in SnGe ₁₂, Sn occupies a peripheral position in SnGe₁₂ (a) structure and occupies a core position in SnGe₁₂ (b) and SnGe₁₂ (c) structures. Compared to other isomers, SnGe₁₂ (a) structure has the highest stability. For charged SnGe₁₂^±, we obtained the same structures with the same symmetry C₁ and Sn–Ge bond length for cation and anion clusters of 3.078 and 3.029 ?, respectively. Two stacked structures (a) and (b) and hexagonal prism (c) are optimized for SnGe₁₃. The calculated results show that the SnGe₁₃ (a) isomer is the most stable geometry and have the same form with pure germanium Ge₁₄. The stacked structure is also stable for SnGe₁₃^±. SnGe₁₄ (a) structure, where the Sn atom is located at the surface, is similar to Ge₁₅ structure. Two other isomers (b) and (c), with Sn atom at the center of the structure, are also found for SnGe₁₄. The same structure with C_s symmetry, have been obtained for SnGe₁₄⁺ and SnGe₁₄^–. SnGe₁₅ have four stable structures. The most stable one is the SnGe₁₅ (a) with C_s symmetry, which is obtained by substitution of Ge atom by Sn atom in the Ge₁₆ cluster. The other isomers SnGe₁₅ (b, c, d) have spherical structures with Sn atom in surface for (b) and in a core position for (c, d). The SnGe₁₅ (a) structure is conserved for the cationic SnGe₁₅⁺ cluster. SnGe₁₆ and SnGe₁₆^± have the same stable structure Ge₁₇ (a) with C₁ symmetry, SnGe₁₆ has two other isomers. Three SnGe₁₇ isomers are shown in Fig. 2. The SnGe₁₇ (a) isomer has the same form of Ge₁₈ and it is the most stable structure. The SnGe₁₇ (b) cluster is similar spherical structure and SnGe₁₇ (c) cluster is spherical structure, Sn atom is located in the centre position in both structures. The same structure as SnGe₁₇ (a) is found for cationic SnGe₁₇⁺ with C₁ symmetry. Different forms of anionic SnGe₁₇^– cluster are obtained.

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Figure2.
(Color online) Ground state structures and their corresponding isomers for SnGe_n (n = 1–17) clusters.

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Figure1.
(Color online) Ground state structures and their isomers for Ge_n+1 (n = 1–17) clusters.

3.2
Electronic properties

In order to explore the relative stability of different species, we calculate the binding energy per atom for the Ge_n₊₁ and SnGe_n^(0±1) (n = 1–17) clusters. The binding energy per atom (E_b/atom) is considered, in cluster science, as a sensitive quantity that reflects the relative stability. The different values of E_b/atom for the lowest energy structures are calculated as:

$${E_{
m{b}}}left( {{
m{G}}{{
m{e}}_n}_{ + 1}}
ight){
m{ }} = {
m{ }}left[ {left( {n + 1}
ight)Eleft( {{
m{Ge}}}
ight) - Eleft( {{
m{G}}{{
m{e}}_n}_{ + 1}}
ight)}
ight]/left( {n + 1}
ight),$$

(1)

$${E_{
m{b}}}left( {{
m{SnG}}{{
m{e}}_n}}
ight){
m{ }} = {
m{ }}[nEleft( {{
m{Ge}}}
ight){
m{ }} + Eleft( {{
m{Sn}}}
ight){
m{ }} - E({
m{SnG}}{{
m{e}}_n} )] /left( {n + 1}
ight),$$

(2)

$$begin{split}{E_{
m{b}}}left( {{
m{SnG}}{{
m{e}}_n}^{ - 1}}
ight) = &[left( {n - 1}
ight)Eleft( {{
m{Ge}}}
ight) + Eleft( {{
m{Sn}}}
ight) &+ Eleft( {{
m{G}}{{
m{e}}^{ - 1}}}
ight) - Eleft( {{
m{SnG}}{{
m{e}}_n}^{ - 1}}
ight)]/left( {n + 1}
ight),end{split}$$

(3)

$${E_{
m{b}}}left( {{
m{SnG}}{{
m{e}}_n}^{ + 1}}
ight){
m{ }} =left[ {
m{ }}nEleft( {{
m{Ge}}}
ight){
m{ }} + Eleft( {{
m{S}}{{
m{n}}^{ + 1}}}
ight){
m{ }} - Eleft( {{
m{SnG}}{{
m{e}}_n}^{ + 1}}
ight)
ight]/left( {n + 1}
ight).$$

(4)

where E(Ge), E(Sn), E(Ge^-1) and E(Sn⁺¹) are the total energy of free Ge, Sn, Ge^– and Sn⁺ atoms, respectively. E(Ge_n₊₁) is the total energy of Ge_n₊₁, E(SnGe_n) is the total energy of SnGe_n and E(SnGe_n^±) is the total energy cationic and anionic SnGe_n^± clusters. The obtained results of binding energies per atom for pure Ge and neutral and charged tin-doped germanium clusters are given in Tables 2, 3 and 5, respectively and their variations versus the cluster size are plotted in Fig. 4. For all species, the binding energy per atom increases with the cluster size. This behavior implies that the cluster stability is enhanced whenever the cluster size increased. The graphs show a thermodynamic instability of smaller clusters. In the case of SnGe_n clusters, the binding energy per atom increases significant increasing in the average binding energies of the clusters of small size range (n < 9), which is due to the rapidly from 1.19 eV/atom in SnGe cluster to 2.77 eV/atom in SnGe ₉. For clusters of size n > 9, we can see a slow increasing in Eb to reach the saturation plateau at 2.8 eV/atom. From Fig. 4, we also observe that there is a high competition in the evolution of the binding energies between SnGe_n and its corresponding Ge_n₊₁ cluster. This means that the Sn atom haven’t an immediate and direct effect on the relative stability of pure germanium clusters.

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Figure4.
(Color online) Size dependence of the binding energies of Ge_n+1 and SnGe_n^{(0, ±1)} (n = 1–17) clusters.

The anionic clusters exhibit high stabilities compared to the others’ cationic and neutral ones. Local peaks are observed in the curve of binding energies of Ge_n₊₁ and SnGe_n for n = 6, n = 9 and n = 13. This means that the corresponding clusters are more stable than their neighbors.

On the other hand, we calculated the fragmentation energy which is considered as a good criterion for predicting the relative stability of the clusters for spontaneous fragmentation. In this works, the size dependence of the fragmentation energies (E_F) for Ge_n₊₁ and SnGe_n^{(0, ±1)} clusters are studied. The fragmentation energy values for different clusters can be calculated by using the following formulas:

$${E_{
m{F}}}left( {{
m{G}}{{
m{e}}_n}_{ + 1}}
ight){
m{ }} = Eleft( {{
m{G}}{{
m{e}}_n}}
ight) + Eleft( {{
m{Ge}}}
ight) - Eleft( {{
m{G}}{{
m{e}}_n}_{ + 1}}
ight),$$

(5)

$$begin{split}{E_{
m{F}}}left( {{
m{SnG}}{{
m{e}}_n}^{(0 pm 1)}}
ight) =& Eleft( {{
m{SnG}}{{
m{e}}_n}^{(0 pm 1)}}
ight)& + Eleft( {{
m{Ge}}}
ight) - Eleft( {{
m{SnG}}{{
m{e}}_n}^{(0 pm 1)}}
ight).end{split}$$

(6)

where E(Ge), E(Ge_n₊₁) and E(SnGe_n^(0±1)) are the total energy of free Ge atom, Ge_n₊₁ cluster and SnGe_n^(0±1) cluster respectively. Based on the above formulas, the calculated fragmentation energy values and their evolution with the cluster size are shown in Fig. 5. We observe that there are oscillating behaviors with an odd–even decreasing tendency in the evolution of fragmentation energy for all the species. Two parts can be distinguished for Ge_n₊₁ and SnGe_n clusters. For n < 9, the clusters with n even values are more stable than n odd values. However, for n > 9, the clusters with n odd values are more stable than n even values. This confirms that the size n = 9 is the stability transition size of these two species. Local maxima of the fragmentation energy of Ge_n₊₁, SnGe_n, SnGe_n⁺ and SnGe_n^? clusters appear at Ge_{4, 10, 14}, SnGe_{3, 9, 13, 16}, SnGe⁺_{3, 5, 13, 16} and SnGe^?_{4, 6, 8, 13}, respectively, which indicates that these clusters are more stable than their neighbors.

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Figure5.
(Color online) Size dependence of the fragmentation energy of Ge_n+1 and SnGe_n^{(0, ± 1)} (n = 1–17) clusters.

In order to analyze the relative stability of Ge_n₊₁ and SnGe_n^(0±1) clusters, we calculate second order difference of total energy (Δ₂E). For the ground state structures the second difference energies for all species are defined by:

$${Delta _2}Eleft( {{
m{G}}{{
m{e}}_n}_{ + 1}}
ight) = Eleft( {{
m{G}}{{
m{e}}_n}_{ + 2}}
ight){
m{ }} + Eleft( {{
m{G}}{{
m{e}}_n}}
ight){
m{ }} - {
m{ }}2Eleft( {{
m{G}}{{
m{e}}_n}_{ + 1}}
ight),$$

(7)

$$begin{split}{Delta _2}Eleft( {{
m{G}}{{
m{e}}_n}_{ + 1}}
ight) =& Eleft( {{
m{SnG}}{{
m{e}}_{n{
m{ + }}1}}^{(0 pm 1)}}
ight) &+ Eleft( {{
m{SnG}}{{
m{e}}_{n - 1}}^{(0 pm 1)}}
ight) - 2Eleft( {{
m{SnG}}{{
m{e}}_n}^{(0 pm 1)}}
ight),end{split}$$

(8)

where E is the total energy of corresponding cluster. It is very known in cluster physics that the small systems with positive value of Δ₂E are more stable than the systems with negative value of Δ₂E. The calculated values of Δ₂E and their evolution as function of the n size for Ge_n₊₁ and SnGe_n^(0±1) are shown in Fig. 6. From Fig. 6, the very pronounced peaks are observed at the sizes n = 1, 5, 7, 10, 12, 14 for pure Ge_n₊₁ and SnGe_n clusters, which demonstrate that the clusters corresponding for these sizes are more stable compared to their neighbors for both species. The explanation of this behavior can be given with a direct relationship between the stability of Sn-doped germanium and their corresponding pure Ge_n₊₁ with the same size. Except for the two size n = 7 and n = 10, we observe that the structure SnGe_n clusters kept unchanged after the encapsulation of Sn atom in pure germanium cage. We also observe that the local peaks at sizes n = 2, 4, 12 and n = 7, 10, 11 for SnGe_n⁺ and SnGe_n^? clusters, respectively indicating that the corresponding clusters have a higher relative stability compared to neighboring clusters.

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Figure6.
(Color online) Second energy differences for Ge_n+1 and SnGe_n^{(0, ± 1)} (n = 1–17) clusters.

In order to understand the chemical reactivity and kinetic stability of the clusters, we calculate the energy differences between the highest occupied and lowest unoccupied molecular orbital (HOMO-LUMO) of the Ge_n₊₁ and SnGe_n^{(0, ±1)} clusters. In Fig. 7, we show the evolution of the HOMO-LUMO gaps as a function of the size for the ground states of different species studied in this piper. We observe a high oscillation of HOMO-LUMO gaps values for the very small size clusters (n < 4). From n = 4 this behavior decreases when the size increases. Many local maxima are observed at n = 2, 4, 8, 9, 11, 13, 15, 17 for both pure and doped germanium clusters, which indicates that these clusters are less reactive than their neighbors and have high chemical stability. The HOMO-LUMO gap of SnGe₃ is lower than that of Ge₄, which indicates that the doping Sn atom enhance the chemical activity of the cluster. We note that the cluster of tin-doped germanium at n = 6 have a large value of HOMO-LUMO gap, which indicates that SnGe₆ is the most chemically stable structure and can be used as the building blocks in many new nanomaterials with specific properties. The ionization of SnGe_n leads to a considerable reduction of the HOMO-LUMO gaps compared to the case of neutral clusters. This indicates that the chemical activity of charged SnGe_n^± clusters is higher than that of neutral SnGe_n clusters. Moreover, the clusters of Ge₂, SnGe_{1, 3} and SnGe^–_{12, 13, 16, 17} have small values of HOMO-LUMO gaps, which indicates that they have partially metallic character.

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Figure7.
(Color online) Size dependence of the HOMO-LUMO gaps of Ge_n+1 and SnGe_n^{(0, ± 1)} (n = 1–17) clusters.

In cluster physics, the ionization potential and electron affinity are considered as suitable parameters to determine the stability of clusters and reflecting the variation of the electronic structure with the size. In order to determine the required energies to add or remove an electron from the lowest energy structures of neutral SnGe_n clusters, we have calculated the adiabatic electron affinity and adiabatic ionization potential, taking in the account the structural relaxation. The vertical ionization potential and vertical electron affinity are respectively the energy required to remove or add an electron on the neutral clusters without structural relaxation.

The ionization potential (IP) is an important character in understanding electronic properties of clusters and giving more information about clusters metallic character. From Fig. 8(a), we can see that VIP and AIP values decrease as the cluster size the increase. It is well known that when the IP is small, the cluster will be closer to a metallic character. This means that the clusters of SnGe_n with size more than 9 atoms exhibit a high metallic character. The smallest VIP and AIP values are observed in SnGe_{14, 16} clusters, indicating that these clusters are more readily ionized than the others. The highest value of SnGe₄ VIP can be explained by its related symmetry. The calculated AEA and VEA of SnGe_n clusters are presented in Fig. 8(b). We can see that the electron affinity increases whenever the cluster size increases. Consequently, the SnGe_n clusters with large sizes will liberate more energy when they capture one electron. We also observe some maximal local peaks in this graph at n = 8, 12, 14 for AEA and n = 7, 9, 15 for VEA, which means that these clusters are energetically less stable than their neighbors.

onerror="this.onerror=null;this.src='http://www.jos.ac.cn/fileBDTXB/journal/article/jos/2018/4/PIC/17070037-8.jpg'"
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Figure8.
(Color online) (a) Vartical (VIP) and adiabatic (AIP) ionization potential; (b) Vartical (VEA) and adiabatic (AEA) electron affinity, (c) chemical hardness for SnGe_n (n = 1 – 17) clusters.

On the other hand, the chemical hardness (η) of cluster can be examined in order to understand its chemical stability. A large value of η indicates that corresponding clusters are less reactive^{[43, 44]}. For the ground state structures, the chemical hardness is evaluated by using the following formula:

$${
m{eta = VIP - VEA}}.$$

(9)

The obtained results for the most stable structures of SnGe_n clusters are listed in Table 3 and plotted in Fig. 8(c) as a function of the size n. We observe that the chemical hardness of SnGe_n clusters decreases when the size increases. This means that clusters with large size are less reactive and more stable than clusters with small size. We can also see from graph that the SnGe₄ cluster has the highest chemical hardness value, which indicates that this cluster is more stable than its neighbors. In addition, other local peaks at n = 2, 6 and 8 for SnGe_n are less reactive than the other clusters.

3.3
Magnetic properties

One of the most important properties that make a special characteristic of these small clusters is their magnetic behavior. Indeed, we can observe very small clusters with very specific magnetic response that can have many important applications in nanotechnologies.

In our case, the magnetic properties are evaluated by the assessment of the total spin magnetic moment for each cluster size. It is defined as the difference between the total Mullikan charge populations for electrons with spin up and electrons with spin down. The calculated total spin magnetic moment values of SnGe_n clusters are reported in Table 3. We observe that all the ground states of SnGe_n clusters are generally nonmagnetic structures, except the case of SnGe₁, which have a total spin magnetic moment of 2μ_b. In order to explore this specific case we plot the total and the partial densities of states for SnGe₁ cluster (DOS and PDOS) (Fig. 9). We observe, from the figure, that the total spin magnetic moment in SnGe monomer is mainly due to the 4p and 5p orbital of Ge and Sn atoms, respectively. In the case of the anionic and cationic SnGe_n clusters, the obtained total spin magnetic moments have been reported in Table 4. We observe that their total spin magnetic moments of ionized structures are different compared to those obtained in neutral structures and they are generally equal to 1μ_b. This means that the charge of the systems can affect the magnetic response of the tin doped germanium clusters.

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Figure9.
(Color online) The total density of states (DOS) for SnGe monomer and the projected density of states (PDOS) for Sn and Ge atoms in SnGe.

4.
Conclusion

In this work, the geometry, stability, electronic and magnetic properties of the neutral and charged SnGe_n clusters have been systematically investigated by the first principles DFT calculations. All the Ge_n₊₁ and SnGe_n^{(0, ±1)} clusters adopt planar structures for very small size and near spherical and spherical compact structures as the size increase. In SnGe_n^{(0, ±1)} clusters, the Sn atom prefers the peripheral position when n < 12 and occupies a core position for n > 12. The structure SnGe _n clusters kept unchanged after the encapsulation of Sn atom in pure germanium cage. For Ge_n₊₁ and SnGe_n^{(0, ±1)} clusters, the binding energy per atom increases with the increasing of the size. The strong increasing in stability of the small clusters is related to the thermodynamic instability of smaller systems. The fragmentation energy calculations indicates that the clusters Ge_{4, 10, 14} and SnGe_{3, 9, 13, 16} and SnGe⁺_{3, 5, 13, 16} are more stable than their neighbors. The second energy difference analysis shows that the clusters of Ge_n₊₁ and SnGe_n clusters at n = 1, 5, 7, 10, 12, 14 are more stable. The HOMO-LUMO gaps of all systems showed a decreasing behavior with the increasing of the size from n = 4 and the charged clusters showed a considerable reduction of the HOMO-LUMO gaps compared to the case of neutral ones. The ionization potentials and electron affinity calculations showed that the clusters of SnGe_n with a size more than 9 atoms exhibit high metallic character and will liberate more energy when they capture one electron. However, the chemical hardness of SnGe_n showed that the clusters with large size are less reactive and more stable. All of the ground states of SnGe_n clusters are nonmagnetic structures, except the case of SnGe₁. Their total spin magnetic moment is mainly due to the 4p and 5p orbital of Ge and Sn atoms, respectively. All these specific properties make the SnGe_n clusters good potential candidates for many eventual applications in nanotechnologies. We have to mention that it is very interesting for the future to calculate the frequencies of vibration in order to obtain the best and more stable isomers.

Acknowledgements

The authors are thankful to the SIESTA group for providing their computational code and greatly acknowledge for LENREZA Laboratory, Ouargla University, Algeria.