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Growth behavior and electronic properties of Ge<sub><i>n</i> + 1</sub> and A

本站小编 Free考研考试/2022-01-01



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[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 Gen cages can be stabilized through the encapsulation of guest atom inside the cage. This was seen before in Sin cages. Matthias Brack et al.[11] presented the global minimum of CuGe10+ clusters as a magic number along with D4d symmetry. Han et al.[12] have presented a theoretical investigation of very small Gen (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 Gen cages having from 16 to 20 atoms. Wang and Han[14] have investigated CuGen (n = 2–13) clusters and shown that Cu doping can decreaseits binding energies, and so, the stability of Gen + 1 clusters. Zhao and Wang have studied in 2009 Mn-doped Gen clusters[15] and shown that Mn dopant can contribute to the stability increase of Gen+1 clusters. Jaiswal and Kumarusing studied the atomic and electronic structures of both neutral and negatively charged ZrGen (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 VGen (n = 1–19) clusters and found that V atom in VGen can make the stability stronger starting from n = 7. More recently, Mahtout et al.[17] have studied the structural, energetic, and electronic properties of MGen 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 Gen + 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 Gen + 1, neutral cationic and anionic SnGen (n = 1–17) clusters. They found that the Sn atom occupied a peripheral position for SnGen 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 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)[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 ?, 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/?.



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.






Symmetry Our work Bibliographic date[24-40]
a (?) Eb (eV) VIP (eV) VEA (eV) a (?) Eb (eV) VIP (eV) VEA (eV)
Ge2 2.503 1.445 7.362 1.473 2.450 1.446 7.844 1.900
2.413 ~1.430 7.627 1.751
2.540 1.620 7.934 1.549
2.610 1.812
2.570 ~1.350
2.420 1.410
2.440 1.320
2.421 1.230
2.625 1.280
Ge3 2.370 2.110 8.024 1.306 2.546 2.059 7.804 2.200
2.400 2.240
2.476 2.150
2.040
1.860
As2 2.143 1.686 9.677 0.132 2.189 1.763
2.103
2.192





Table1.
Averaged bond length a, binding energy Eb, vertical ionization potential VIP, and vertical electron affinity VEA for Ge2, Ge3 and As2.



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Symmetry Our work Bibliographic date[24-40]
a (?) Eb (eV) VIP (eV) VEA (eV) a (?) Eb (eV) VIP (eV) VEA (eV)
Ge2 2.503 1.445 7.362 1.473 2.450 1.446 7.844 1.900
2.413 ~1.430 7.627 1.751
2.540 1.620 7.934 1.549
2.610 1.812
2.570 ~1.350
2.420 1.410
2.440 1.320
2.421 1.230
2.625 1.280
Ge3 2.370 2.110 8.024 1.306 2.546 2.059 7.804 2.200
2.400 2.240
2.476 2.150
2.040
1.860
As2 2.143 1.686 9.677 0.132 2.189 1.763
2.103
2.192








Continued from Table 2
Cluster size (n) Symmetry Eb (eV/atom) ΔE (eV) VEA (eV) VIP (eV) η (eV) aGe–Ge (?)
15 C2h 3.027 1.436 6.840 2.308 4.532 2.715
C1 3.020 0.883 6.555 2.494 4.061 2.755
C2 3.034 1.364 6.755 2.262 4.493 2.808
C2h 3.095 1.393 7.548 1.753 5.795 2.780
16 Cs 3.050 1.103 6.588 2.343 4.245 2.781
C1 3.043 1.254 6.696 2.428 4.268 2.772
C1 3.038 1.022 6.616 2.500 4.116 2.762
Cs 3.077 0.858 7.075 1.716 5.359 2.823
17 C2 3.014 0.864 6.493 2.590 3.903 2.869
Cs 3.013 0.948 6.537 2.537 4.000 2.790
Cs 3.009 0.555 6.400 2.815 3.585 2.782
C 3.062 1.452 7.152 1.486 5.666 2.729
18 C1 3.053 0.966 6.473 2.477 3.996 2.843
C1 3.019 0.779 6.516 2.728 3.788 2.771
C1 3.019 0.780 6.517 2.727 3.790 2.771
19 C1 3.046 0.828 6.387 2.596 3.791 2.735
C1 3.033 0.962 6.365 2.455 3.910 2.768
C1 3.001 0.828 6.418 2.650 3.768 2.781
20 C1 3.041 0.743 6.372 2.679 3.693 2.769
C1 3.061 3.061 6.403 2.845 3.558 2.735
C1 3.050 0.559 6.182 2.734 3.448 2.761






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Continued from Table 2
Cluster size (n) Symmetry Eb (eV/atom) ΔE (eV) VEA (eV) VIP (eV) η (eV) aGe–Ge (?)
15 C2h 3.027 1.436 6.840 2.308 4.532 2.715
C1 3.020 0.883 6.555 2.494 4.061 2.755
C2 3.034 1.364 6.755 2.262 4.493 2.808
C2h 3.095 1.393 7.548 1.753 5.795 2.780
16 Cs 3.050 1.103 6.588 2.343 4.245 2.781
C1 3.043 1.254 6.696 2.428 4.268 2.772
C1 3.038 1.022 6.616 2.500 4.116 2.762
Cs 3.077 0.858 7.075 1.716 5.359 2.823
17 C2 3.014 0.864 6.493 2.590 3.903 2.869
Cs 3.013 0.948 6.537 2.537 4.000 2.790
Cs 3.009 0.555 6.400 2.815 3.585 2.782
C 3.062 1.452 7.152 1.486 5.666 2.729
18 C1 3.053 0.966 6.473 2.477 3.996 2.843
C1 3.019 0.779 6.516 2.728 3.788 2.771
C1 3.019 0.780 6.517 2.727 3.790 2.771
19 C1 3.046 0.828 6.387 2.596 3.791 2.735
C1 3.033 0.962 6.365 2.455 3.910 2.768
C1 3.001 0.828 6.418 2.650 3.768 2.781
20 C1 3.041 0.743 6.372 2.679 3.693 2.769
C1 3.061 3.061 6.403 2.845 3.558 2.735
C1 3.050 0.559 6.182 2.734 3.448 2.761






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[10, 14, 18, 24, 25] using DFT different calculations. The triangular geometry with C2v symmetry is found to be the lowest-energy structure for Ge3. The lowest-energy isomer of the tetramer Ge4 has D2h symmetry in line with previous findings[15, 18, 20, 26, 27]. The most favorable isomer for Ge5 cluster reveals a triangular bipyramid disposition with D3h symmetry, which is also in line with the previous data[10, 14, 24, 26]. The lowest-energy Ge6 cluster has bicapped quadrilateral structure with C2v symmetry, in good agreement with the previous data[10, 18, 26]. For Ge7 cluster, the most stable structure reveals a pentagonal bipyramid of D5h 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 C2v symmetry, as obtained in earlier works[20, 28]. Ge9 is a capped pentagonal bipyramid structure of C2v symmetry. The most favorable isomer of Ge10 cluster is a capped pentagonal basis structure and has C3v symmetry. The lowest-energy Ge11 cluster shows a compact near spherical geometry of Cs symmetry. As per Ge12 and Ge13 clusters, prolate-type structure with C2v symmetry was always preferred. The shape of Ge14 is a prolate structure with Cs symmetry. For Ge15 and Ge16 clusters, prolate-type geometry was obtained with C2v and C2h symmetries, respectively. The lowest-energy isomer for Ge17 possesses a near spherical geometry of Cs symmetry. From Ge18 to Ge21 clusters, prolate-type shape with C1 symmetry was always preferred.






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Figure1.
(Color online) Most favorable structures together with their corresponding isomers for Gen + 1 (n = 1–20) clusters.






Cluster size (n) Symmetry Eb (eV/atom) ΔE (eV) VEA (eV) VIP (eV) η (eV) aGe–Ge (?)
1 D∞h 1.445 0.265 7.362 1.473 5.889 2.503
2 D∞h 2.048 1.272 7.557 1.476 6.081 2.339
C2v 2.109 1.543 8.024 1.305 6.719 2.370
C2v 2.110 1.543 8.024 1.306 6.718 2.370
3 D4h 2.228 0.455 6.834 1.599 5.235 2.550
D2h 2.556 1.180 7.734 1.758 5.976 2.576
D2h 2.557 1.179 7.733 1.758 5.976 2.597
C3v 2.078 0.660 7.056 1.717 5.339 2.479
D∞h 2.061 1.169 7.114 2.102 5.012 2.346
4 C2 2.518 1.089 6.911 2.149 4.762 2.606
C2v 2.450 1.026 6.766 2.207 4.559 2.564
C2v 2.499 0.808 6.774 2.070 4.704 2.618
C2v 2.504 0.577 7.588 2.560 5.028 2.775
D3h 2.707 2.036 8.672 0.218 8.454 2.547
5 D4h 2.847 1.998 7.777 1.372 6.405 2.782
C2h 2.668 1.236 7.142 1.820 5.322 2.760
C2 2.672 1.386 7.172 1.790 5.382 2.606
C2v 2.848 1.957 7.742 1.359 6.383 2.710
6 D3v 2.877 2.350 7.460 1.064 6.396 2.734
C1 2.687 0.164 7.304 1.673 5.631 2.647
C2 2.843 1.170 7.034 1.749 5.285 2.754
D5h 2.974 1.836 7.875 1.774 6.101 2.747
C2 2.843 1.170 7.034 1.748 5.286 2.754
Cs 2.617 0.755 6.620 2.102 4.518 2.701
7 C2v 2.866 0.980 7.216 2.232 4.984 2.776
C2 2.735 0.977 6.937 2.194 4.743 2.658
Cs 2.739 0.708 6.556 2.212 4.344 2.730
Cs 2.422 0.431 6.194 2.805 3.389 2.651
8 C2v 2.985 1.654 7.126 1.570 5.556 2.782
C1 2.700 1.066 6.849 2.394 4.455 2.570
D3d 2.574 0.227 6.288 2.271 4.017 2.798
C3v 2.827 1.102 7.068 2.313 4.755 2.686
9 Cs 2.848 1.459 6.462 2.146 4.316 2.742
Cs 3.002 1.753 7.137 1.662 5.475 2.779
C2v 2.968 1.379 7.070 1.981 5.089 2.785
C3v 3.082 1.812 7.432 1.857 5.575 2.775
C1 2.953 1.295 6.900 1.913 4.987 2.738
Cs 3.013 1.015 7.152 2.393 4.759 2.794
Cs 3.003 1.547 7.410 1.383 6.027 2.771
10 C3v 2.792 1.063 6.402 2.364 4.038 2.714
D4h 2.715 0.793 6.326 1.975 4.351 2.816
Cs 2.907 0.962 6.605 2.084 4.521 2.748
Cs 2.892 0.990 6.820 2.318 4.502 2.724
Cs 3.029 1.258 7.107 1.332 5.775 2.770
11 Cs 2.936 0.803 6.852 2.547 4.305 2.798
C1 2.955 0.916 6.759 2.345 4.414 2.784
C1 2.964 1.059 6.696 2.175 4.521 2.801
Cs 2.979 0.370 6.656 2.780 3.876 2.793
C2v 3.032 1.793 7.402 1.258 6.144 2.744
12 C2v 3.050 1.181 8.243 1.290 6.953 2.760
C2 3.007 1.153 1.153 2.390 4.553 2.792
C1 3.015 0.945 6.785 2.412 4.373 2.835
Cs 2.990 1.130 6.608 2.102 4.506 2.769
C1 3.015 0.944 7.302 1.779 5.523 2.831
13 C3v 2.922 0.987 5.967 2.633 3.334 2.696
Cs 3.026 1.107 6.625 2.179 4.446 2.700
Oh 2.977 1.007 6.915 3.095 3.820 2.659
C1 2.986 1.036 6.471 2.127 4.344 2.784
Cs 3.092 1.628 7.486 1.539 5.947 2.797
14 D3d 2.864 0.495 6.726 2.977 3.749 2.666
C1 2.999 1.234 6.800 2.311 4.489 2.829
Cs 2.959 0.941 6.553 2.393 4.160 2.785
C1 3.022 1.149 6.579 2.178 4.401 2.812
C1 3.023 1.156 6.927 2.468 4.459 2.786
C2v 3.082 0.899 7.339 1.851 5.488 2.814





Table2.
Group of symmetry, Eb, ΔE, VIP, VEA, η, and aGe–Ge for pure Gen + 1 (n = 1 – 20) clusters.



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Cluster size (n) Symmetry Eb (eV/atom) ΔE (eV) VEA (eV) VIP (eV) η (eV) aGe–Ge (?)
1 D∞h 1.445 0.265 7.362 1.473 5.889 2.503
2 D∞h 2.048 1.272 7.557 1.476 6.081 2.339
C2v 2.109 1.543 8.024 1.305 6.719 2.370
C2v 2.110 1.543 8.024 1.306 6.718 2.370
3 D4h 2.228 0.455 6.834 1.599 5.235 2.550
D2h 2.556 1.180 7.734 1.758 5.976 2.576
D2h 2.557 1.179 7.733 1.758 5.976 2.597
C3v 2.078 0.660 7.056 1.717 5.339 2.479
D∞h 2.061 1.169 7.114 2.102 5.012 2.346
4 C2 2.518 1.089 6.911 2.149 4.762 2.606
C2v 2.450 1.026 6.766 2.207 4.559 2.564
C2v 2.499 0.808 6.774 2.070 4.704 2.618
C2v 2.504 0.577 7.588 2.560 5.028 2.775
D3h 2.707 2.036 8.672 0.218 8.454 2.547
5 D4h 2.847 1.998 7.777 1.372 6.405 2.782
C2h 2.668 1.236 7.142 1.820 5.322 2.760
C2 2.672 1.386 7.172 1.790 5.382 2.606
C2v 2.848 1.957 7.742 1.359 6.383 2.710
6 D3v 2.877 2.350 7.460 1.064 6.396 2.734
C1 2.687 0.164 7.304 1.673 5.631 2.647
C2 2.843 1.170 7.034 1.749 5.285 2.754
D5h 2.974 1.836 7.875 1.774 6.101 2.747
C2 2.843 1.170 7.034 1.748 5.286 2.754
Cs 2.617 0.755 6.620 2.102 4.518 2.701
7 C2v 2.866 0.980 7.216 2.232 4.984 2.776
C2 2.735 0.977 6.937 2.194 4.743 2.658
Cs 2.739 0.708 6.556 2.212 4.344 2.730
Cs 2.422 0.431 6.194 2.805 3.389 2.651
8 C2v 2.985 1.654 7.126 1.570 5.556 2.782
C1 2.700 1.066 6.849 2.394 4.455 2.570
D3d 2.574 0.227 6.288 2.271 4.017 2.798
C3v 2.827 1.102 7.068 2.313 4.755 2.686
9 Cs 2.848 1.459 6.462 2.146 4.316 2.742
Cs 3.002 1.753 7.137 1.662 5.475 2.779
C2v 2.968 1.379 7.070 1.981 5.089 2.785
C3v 3.082 1.812 7.432 1.857 5.575 2.775
C1 2.953 1.295 6.900 1.913 4.987 2.738
Cs 3.013 1.015 7.152 2.393 4.759 2.794
Cs 3.003 1.547 7.410 1.383 6.027 2.771
10 C3v 2.792 1.063 6.402 2.364 4.038 2.714
D4h 2.715 0.793 6.326 1.975 4.351 2.816
Cs 2.907 0.962 6.605 2.084 4.521 2.748
Cs 2.892 0.990 6.820 2.318 4.502 2.724
Cs 3.029 1.258 7.107 1.332 5.775 2.770
11 Cs 2.936 0.803 6.852 2.547 4.305 2.798
C1 2.955 0.916 6.759 2.345 4.414 2.784
C1 2.964 1.059 6.696 2.175 4.521 2.801
Cs 2.979 0.370 6.656 2.780 3.876 2.793
C2v 3.032 1.793 7.402 1.258 6.144 2.744
12 C2v 3.050 1.181 8.243 1.290 6.953 2.760
C2 3.007 1.153 1.153 2.390 4.553 2.792
C1 3.015 0.945 6.785 2.412 4.373 2.835
Cs 2.990 1.130 6.608 2.102 4.506 2.769
C1 3.015 0.944 7.302 1.779 5.523 2.831
13 C3v 2.922 0.987 5.967 2.633 3.334 2.696
Cs 3.026 1.107 6.625 2.179 4.446 2.700
Oh 2.977 1.007 6.915 3.095 3.820 2.659
C1 2.986 1.036 6.471 2.127 4.344 2.784
Cs 3.092 1.628 7.486 1.539 5.947 2.797
14 D3d 2.864 0.495 6.726 2.977 3.749 2.666
C1 2.999 1.234 6.800 2.311 4.489 2.829
Cs 2.959 0.941 6.553 2.393 4.160 2.785
C1 3.022 1.149 6.579 2.178 4.401 2.812
C1 3.023 1.156 6.927 2.468 4.459 2.786
C2v 3.082 0.899 7.339 1.851 5.488 2.814





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 ? and one Ge–Ge bond of 2.775 ?. The As–Ge bond distance of 0.09 ? 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 ?, 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 ?, and an average Ge–Ge bond distance of 2.807 ?. The lowest energy isomer for AsGe6 cluster is a structure with C2v point group symmetry, As–Ge bond length of 2.704 ?, and an average Ge–Ge bond distance of 2.786 ?. 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.






Cluster size (n) Symmetry Eb (eV/atom) ΔE (eV) VEA (eV) VIP (eV) η (eV) aGe–Ge (?) aAs–Ge (?)
1 (a)C∞v1 1.426 0.171 2.175 8.142 5.967 - 2.350
2 (a)C2v 2.139 1.110 0.969 7.425 6.456 2.775 2.445
(b)C2v 2.139 1.109 0.778 8.198 7.410 2.775 2.445
3 (a)C2v 2.380 0.589 1.440 6.704 5.264 2.603 2.543
(b)C2v 2.418 1.266 1.726 7.377 5.651 2.661 2.473
4 (a)C2v 2.575 0.208 0.451 8.640 8.189 2.738 2.661
(b)C2v 2.641 0.917 0.124 8.809 8.685 2.692 2.734
5 (a)C4v 2.705 1.116 0.928 6.657 5.729 2.807 2.679
(b)C2v 2.702 1.058 1.088 6.951 5.863 2.782 2.706
6 (a)C2v 2.837 1.279 1.632 6.614 4.982 2.786 2.703
(b)C2v 2.837 1.278 1.632 6.615 4.983 2.786 2.704
7 (a)Cs 2.814 0.503 2.060 7.179 5.119 2.788 2.635
(b)C3v 2.835 0.743 1.701 7.022 5.321 2.821 2.567
8 (a)Cs 2.908 0.571 1.279 6.661 5.382 2.808 2.678
(b)Cs 2.909 0.571 1.279 6.661 5.382 2.808 2.678
9 (a)Cs 2.977 1.066 1.604 6.271 4.667 2.774 2.819
(b)Cs 2.977 1.064 1.603 6.271 4.668 2.774 2.819
10 (a)C1 2.949 0.759 1.051 6.663 5.612 2.793 2.697
(b)C1 2.949 0.898 0.884 6.953 6.069 2.774 2.716
11 (a)Cs 2.945 1.026 0.783 6.609 5.826 2.740 2.813
(b)Cs 2.940 0.773 1.305 6.481 5.176 2.775 2.596
12 (a)C1 2.993 0.393 1.019 8.052 7.033 2.788 2.693
(b)Cs 3.003 0.396 1.044 7.736 6.692 2.814 2.667
13 (a)C1 3.016 0.696 1.518 6.578 5.060 2.826 2.606
(b)C1 3.016 0.697 1.520 6.574 5.054 2.780 2.607
14 (a)C1 3.042 0.581 1.248 7.125 5.877 2.821 2.681
(b)Cs 3.046 0.797 1.174 6.946 5.772 2.828 2.783
15 (a)Cs 3.045 0.782 1.482 6.984 5.502 2.801 2.736
(b)C1 3.043 0.846 1.300 6.917 5.617 2.817 2.719
16 (a)C1 3.045 0.538 1.394 6.977 5.583 2.813 2.595
(b)Cs 3.035 0.810 1.547 6.823 5.276 2.830 2.593
17 (a)C1 3.031 0.623 0.763 7.031 6.268 2.766 2.566
(b)C1 3.030 0.390 0.453 7.310 6.857 2.747 2.556
18 (a)C1 3.017 0.627 1.867 6.346 4.479 2.777 2.629
(b)C1 3.024 0.594 2.000 6.361 4.361 2.803 2.592
19 (a)C1 3.023 0.559 1.916 6.098 4.182 2.769 2.581
(b)C1 3.029 0.519 1.667 6.834 5.167 2.732 2.580
20 (a)C1 3.065 0.667 1.961 6.558 4.597 2.739 2.573
(b)C1 3.059 0.579 1.870 6.937 5.067 2.745 2.601





Table3.
Group of symmetry, Eb, ΔE, VEA, VIP, η, and aGe–Ge, aAs–Ge for AsGen (n = 1–20) clusters.



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Cluster size (n) Symmetry Eb (eV/atom) ΔE (eV) VEA (eV) VIP (eV) η (eV) aGe–Ge (?) aAs–Ge (?)
1 (a)C∞v1 1.426 0.171 2.175 8.142 5.967 - 2.350
2 (a)C2v 2.139 1.110 0.969 7.425 6.456 2.775 2.445
(b)C2v 2.139 1.109 0.778 8.198 7.410 2.775 2.445
3 (a)C2v 2.380 0.589 1.440 6.704 5.264 2.603 2.543
(b)C2v 2.418 1.266 1.726 7.377 5.651 2.661 2.473
4 (a)C2v 2.575 0.208 0.451 8.640 8.189 2.738 2.661
(b)C2v 2.641 0.917 0.124 8.809 8.685 2.692 2.734
5 (a)C4v 2.705 1.116 0.928 6.657 5.729 2.807 2.679
(b)C2v 2.702 1.058 1.088 6.951 5.863 2.782 2.706
6 (a)C2v 2.837 1.279 1.632 6.614 4.982 2.786 2.703
(b)C2v 2.837 1.278 1.632 6.615 4.983 2.786 2.704
7 (a)Cs 2.814 0.503 2.060 7.179 5.119 2.788 2.635
(b)C3v 2.835 0.743 1.701 7.022 5.321 2.821 2.567
8 (a)Cs 2.908 0.571 1.279 6.661 5.382 2.808 2.678
(b)Cs 2.909 0.571 1.279 6.661 5.382 2.808 2.678
9 (a)Cs 2.977 1.066 1.604 6.271 4.667 2.774 2.819
(b)Cs 2.977 1.064 1.603 6.271 4.668 2.774 2.819
10 (a)C1 2.949 0.759 1.051 6.663 5.612 2.793 2.697
(b)C1 2.949 0.898 0.884 6.953 6.069 2.774 2.716
11 (a)Cs 2.945 1.026 0.783 6.609 5.826 2.740 2.813
(b)Cs 2.940 0.773 1.305 6.481 5.176 2.775 2.596
12 (a)C1 2.993 0.393 1.019 8.052 7.033 2.788 2.693
(b)Cs 3.003 0.396 1.044 7.736 6.692 2.814 2.667
13 (a)C1 3.016 0.696 1.518 6.578 5.060 2.826 2.606
(b)C1 3.016 0.697 1.520 6.574 5.054 2.780 2.607
14 (a)C1 3.042 0.581 1.248 7.125 5.877 2.821 2.681
(b)Cs 3.046 0.797 1.174 6.946 5.772 2.828 2.783
15 (a)Cs 3.045 0.782 1.482 6.984 5.502 2.801 2.736
(b)C1 3.043 0.846 1.300 6.917 5.617 2.817 2.719
16 (a)C1 3.045 0.538 1.394 6.977 5.583 2.813 2.595
(b)Cs 3.035 0.810 1.547 6.823 5.276 2.830 2.593
17 (a)C1 3.031 0.623 0.763 7.031 6.268 2.766 2.566
(b)C1 3.030 0.390 0.453 7.310 6.857 2.747 2.556
18 (a)C1 3.017 0.627 1.867 6.346 4.479 2.777 2.629
(b)C1 3.024 0.594 2.000 6.361 4.361 2.803 2.592
19 (a)C1 3.023 0.559 1.916 6.098 4.182 2.769 2.581
(b)C1 3.029 0.519 1.667 6.834 5.167 2.732 2.580
20 (a)C1 3.065 0.667 1.961 6.558 4.597 2.739 2.573
(b)C1 3.059 0.579 1.870 6.937 5.067 2.745 2.601








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Figure2.
(Color online) Most favorable structures and their corresponding isomers of AsGen (n = 1–20) clusters.





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.






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Figure3.
(Color online) Evolution of the binding energy per atom for the lowest energy structures of Gen + 1 and AsGen (n = 1–20) clusters as a function of cluster size.





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.






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Figure4.
(Color online) Evolution of the fragmentation energy of Gen + 1 and AsGen (n = 1–20) clusters as a function of cluster size.





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.






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Figure5.
(Color online) Evolution of the second-order difference of energy for the lowest energy structures of Gen + 1 and AsGen (n = 1–20) clusters as a function of cluster size.





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[28]. Fig. 6 reports the size dependence of HOMO–LUMO gap for the most favorable structures of Gen + 1 and AsGen (n = 1–20) clusters. The decreasing behavior with the size is important for Gen + 1 clusters, while is less pronounced for AsGen clusters. Overall, the gaps of AsGen are much lower than those obtained for Gen + 1 clusters, except for n = 3 and 20, The value for AsGen roughly oscillates in between 0.171 and 1.278 eV, which indicates a chemical activity increase of Gen + 1 clusters when doped with As. Doping Gen + 1 cages with an As atom leads to a significant HOMO–LUMO gap reduction in AsGen clusters. This means that the chemical activity of AsGen is higher than that of Gen + 1 clusters and the inserted As atom highlights the metallic character of AsGen 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 Gen + 1 clusters.






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Figure6.
(Color online) Evolution of the HOMO–LUMO gap for the lowest energy structures of Gen + 1 and AsGen (n = 1–20) clusters as a function of cluster size.





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.






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Figure7.
(Color online) Evolution of the vertical ionization potential (VIP) for the lowest energy structures of Gen + 1 and AsGen (n = 1–20) clusters as a function of cluster size.






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Figure8.
(Color online) Evolution of the vertical electron affinity (VEA) for the lowest energy structures of Gen + 1 and AsGen (n = 1–20) clusters as a function of cluster size.





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[39, 41]. The clusters with high values of hardness are less reactive and more stable.






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Figure9.
(Color online) Evolution of the chemical hardness η for the lowest energy structures of Gen + 1 and AsGen (n = 1–20) clusters as a function of cluster size.





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.



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