Study of the Geometric Structures, Electronic and Magnetic Properties of Aluminium-Antimony Alloy Clusters

3.1 Geometrical Structures

The unbiased CALYPSO structure searching method was carried out to get the global minima structures of neutral and anionic Al_nS band pure Al_n+1(n = 1–16) clusters. Using the Gaussian 09 package [33], the lowest-energy structures of neutral and anionic Al_nSb(n = 1–16) clusters, as well as neutral and anionic Al_n+1(n = 1–16) are obtained and displayed in Figure 1. Their symmetry, electronic state, binding energy E_b and HOMO-LUMO energy gap E_gap are listed in Tables 1 and S1 (Supplementary Material). The corresponding lower-lying isomers of neutral and anionic Al_nSb(n = 1–16) clusters are displayed in Figures S1 and S2 (Supplementary Material). Moreover, the configures of the most stable structures with the same size based on PBE0, mPW1PW91 functionals are different from those based on functional B3PW91, which are summarised in Figure S3 (Supplementary Material).

Figure 1 shows that the most stable structures retain planar for the neutral Al_nSb clusters for n ≤ 4, anionic Al_nSb clusters for n ≤ 3, as well as neutral and anionic Al_n+1 clusters for n ≤ 4. When Al atoms n in cluster is >5, the most stable structures of Al_nSb clusters and pure Al_n+1 clusters display a three-dimensional structure. Moreover, one can notice that the most stable structures of the neutral and anionic Al_nSb clusters are almost the same except for 4, 7 and 11. For the neutral and anionic Al_n+1 clusters, their most stable structures are also the same except for 11 and 12. It is worth noting that, whether Al_nSb clusters or pure Al_n+1 clusters, the structural change is very small when an electron is added, which indicates the most stable structures of the anionic Al_nSb and Al_n+1 clusters can be obtained by attaching an electron from the corresponding neutral cluster in its ground state. By comparing the most stable structures of the Al_nSb clusters with those of the corresponding same size Al_n+1 clusters, we also note that the most stable structures are almost the same for the neutral and anionic Al_nSb, Al_n+1 clusters except for 8, 10 and 11, which further indicates that the Sb dopant will not change the structures of pure Al_n+1 clusters except for 8, 10 and 11. Thus we may obtain the most stable structures of the neutral and anionic binary alloy Al-Sb clusters by substituting an Sb atom for an Al atom in the corresponding neutral and anionic Al_n+1 clusters except for a handful of clusters. In the range of 1 ≤ n ≤ 6, the most stable structures of neutral and anionic Al_nSb clusters are consistent with the other theoretical prediction results. This good agreement further verifies that our calculated results are reliable. However, when cluster size n ≥ 7, we have not found the related reports about the neutral and charged Al_nSb clusters. For n = 7, the most stable structures of the neutral and anionic Al₇Sb are similar and can be obtained by replacing an Al atom with an Sb atom in Al₈ cluster. Interestingly, in the range of 8 ≤ n ≤ 11, except for 9, the most stable structures of the neutral and anionic Al_nSb clusters are different from that of the neutral and anionic Al_n+1 clusters, respectively, which further indicates that the impure Sb changes the structural evolution of pure Al_n+1 clusters. And the same size configures of the neutral and anionic pure Al_n+1(n = 8, 10 and 11) are found in low-lying isomers of Al_nSb(n = 8, 11 and 12) and Al_nSb⁻(n = 8, 11 and 12) clusters (seen in Figs. S1 and S2). Moreover, the structural transition from bilayer-like structures to cage-like structures occurs at n = 11 for neutral clusters and n = 12 for anionic clusters. The most stable structures show that Sb atom is inclined to occupy a peripheral position of the Al_n+1 cluster but not an endohedral position, which is similar to other Al-Mg, Al-Ti, Al-As alloy clusters and different compared to Al-Cu alloy cluster [2], [25], [45], [46]. For 12 ≤ n ≤ 16, the most stable structures are the same for neutral or anionic clusters, indicating that the impurity Sb does not change the structural evolution of pure Al_n+1 clusters.

3.2 Electronic Properties and Photoelectron Spectra

The ionisation potentials, electron affinities, and detachment energies are very important quantities that are usually regarded as a signal to characterise the metallic characteristics in metal clusters. The vertical ionisation potentials (VIP) and vertical electron affinities (VEA) are used to describe the related properties of the neutral cluster, which can be estimated as follows:

VIP = E (cation at optimised neutral geometry) – E (optimised neutral)

VEA = E (optimised neutral) – E (anion at optimised neutral geometry)

The vertical detachment energy (VDE) and adiabatic detachment energy (ADE) can be evaluated as follows:

ADE = E (optimised neutral) – E (optimised anion)

VDE = E (neutral at optimised anion geometry) – E (optimised anion)

Figure 2:

The simulated photoelectron spectra of the anionic Al_nSb(n = 1–16) clusters, as well as the experimental photoelectron spectra.

Our calculated results are listed in Table 2. One can see that the calculated VEA, VIP values of the most stable structures for pure Al_n+1 clusters, and ADE, VDE values of the most stable structures for pure Al_n+1⁻ clusters are in good agreement with some experiment results [2], [20], [47], further indicating the B3PW91 functional is suitable to study the pure Al_n clusters or Al-based metal clusters. Moreover, to further ensure the most stable structures of anionic Al_nSb(n = 1–16) clusters via experimental spectra, the photoelectron spectra of anionic Al_nSb(n = 1–16) clusters are simulated by adding the occupied orbital energy relative to the VDE and fitting them with a Lorentz expansion scheme and broadening factor of 0.1 eV as plotted in Figure 2. This simulated method is applied successfully in other metal clusters [3], [4]. In photoelectron spectra, each peak corresponds to a transition from the ground state of the Al_nSb⁻ clusters to the ground state or an excited state of the corresponding Al_nSb clusters, and the location of the first peak is corresponding to the VDE. Moreover, the ADE for the neutralclusters was measured by the corresponding intersection between the baseline and the rising edge of the first peak. The first peak of AlSb⁻ cluster is located about 2.03 eV, which is in good agreement with the position of the first peak of the experimental photoelectron spectrum 2.13 eV. For Al_nSb⁻(n = 2–5) clusters, position of the first peak in our simulated spectra corresponding to 2.50, 2.67, 2.51 and 2.75 eV are also consistent with the experimental results 2.57, 2.27, 2.62 and 2.87 eV, respectively. Moreover, by comparing the simulated and experimental PES, one can see that the position of the intersection between the baseline and the rising edge of the first peak with the same size is almost identical for both the simulated and experimental PES. Furthermore, we also notice that the calculated values of ADE for anionic Al_nSb(n = 1–5) clusters are 1.99, 2.31, 2.00, 2.16 and 2.32 eV which are in good agreement with the experimental results 1.96, 2.39, 1.85, 1.98 and 2.03 eV, respectively [27]. Based on the good agreement, we simulate the photoelectron spectra of Al_nSb⁻(n = 6–16) clusters, the first peak of Al_nSb⁻(n = 6–16) clusters are located about 2.63, 2.84, 2.66, 2.42, 2.76, 3.18, 2.81, 2.96, 3.14, 2.50 and 2.83 eV, respectively. It needs to be verified by experimental or other theoretical workers. The estimated VIP, VEA for neutral Al_nSb(n = 1–5) clusters agree well with other theoretical values [27]. These agreements further confirm that the predicted ground state structures are reliable and real global minima. Thus, the predicted values of VEA, VIP and ADE for neutral and anionic Al_nSb(n = 6–16) clusters are also calculated and listed in Table 2. In order to assess further the performance of B3PW91 functional, the electronic properties of the neutral and anionic Al_nSb and Al_n+1(n = 1–16) are investigated based on functionals PBE0 and mPW1PW91, the results are listed in Tables S2–S5 (Supplementary Material) compared with functional B3Pw91, indicating that B3PW91 functional is also suitable to study the pure Al or Al-metal alloy clusters due to their consistency by means of functionals PBE0, mPW1PW91 and B3PW91.

3.3 Stability Analysis

The relative stabilities of the investigated clusters are discussed based on the binding energy per atom E_b(n), fragmentation energy ΔE(n), second-order differences of the total energy Δ₂E(n), HOMO-LUMO energy gaps, global chemical hardness and DE.

For the neutral and anionic pure Al_n+1 clusters, the E_b(n), ΔE(n), and Δ₂E(n) can be estimated as the following formulas:

For the neutral and anionic Al_nSb clusters, the E_b (n), ΔE(n) and Δ₂E(n) can be written as follows:

Figure 3:

The averaged binding energies E_b, fragmentation energies ΔE, second-order energy differences Δ₂E and HOMO-LUMO energy gaps E_gap vs. the cluster size n for the most stable structures of the neutral and anionic Al_nSb and pure Al_n+1(n = 1–16) clusters.

where E(Al), E(Al^q), E(Al_n^q), E(Al_n−1^q), E(Al_n+1^q), E(Sb^q), E(Al_nSb^q), E(Al_n−1Sb^q), and E(Al_n+1Sb^q) indicate the total energies of the ground state structures of Al, Al^q, Al_n^q, Al_n−1^q, Al_n+1^q, Sb^q, Al_nSb^q, Al_n−1Sb^q, and Al_n+1Sb^q clusters, respectively. The E_b(n), ΔE(n), Δ₂E(n) values of the neutral and anionic pure Al_n+1 clusters and Al_nSb(n = 1–16) clusters are plotted against n in Figure 3. The plots indicate that the binding energy per atom increases rapidly with the increasing cluster size, revealing that the Al_n+1 and Al_nSb(n = 1–16) clusters can continuously gain energy during the cluster evolution process. Moreover, we know that the peaks appeared in Figure 3 represent relatively more stable clusters. Some peaks are found corresponding to Al_n(n = 5, 7, 13, 16), Al_n⁻(n = 7, 9, 13), Al_nSb(n = 3, 5, 9, 15), Al_nSb⁻(n = 2, 5, 8, 12, 14) clusters, respectively, signifying that these clusters are obviously more stable than their neighbouring clusters, respectively. Moreover, the binding energy per atom E_b (n) of neutral and anionic pure Al_n+1 clusters and Al_nSb(n = 1–16) clusters is studied by means of a functional mPW1PW91 and PBE0 which are summarised in Table S7 compared with those based on B3PW91 functional. The consistency of results based on three functionals further indicates that the B3PW91 functional is good for pure Al and Al-antimony alloy clusters.

The HOMO-LUMO energy gap is another sensitive quantity used to reveal the relative stability of the clusters. The energy gap values of the neutral and anionic pure Al_n+1 clusters and Al_nSb(n = 1–16) clusters are also plotted against n in Figure 3. Some peaks are observed which correspond to Al_n(n = 5, 7, 12, 14), Al_n⁻(n = 6, 13), Al_nSb with odd n and Al_nSb⁻(n = 5, 8, 14, 16) clusters, respectively. It is worth noting that Al₃Sb and Al₅Sb present large HOMO-LUMO energy gaps at 2.59 and 2.09 eV, respectively, which are only slightly smaller than that of Al₁₃⁻ (2.74 eV). Thus, it is possible that the Al₃Sb and Al₅Sb are relatively more stable structures as the cluster size is up to 16. Moreover, we also noted that the HOMO-LUMO energy gaps of neutral Al_nSb clusters with even n are smaller than those of Al_n+1, indicating that the impurity Sb atom can improve the chemical reactivity of neutral Al_nSb clusters except Al_nSb(n = 2, 8, 12) clusters. Similarly, the chemical reactivity of Al_nSb⁻ with odd n as well as Al₁₀Sb⁻ and Al₁₂Sb⁻ was enhanced by doping Sb atom except for Al_nSb⁻(n = 3, 7, 11). Moreover, the chemical hardness can be estimated in terms of the finite difference approximation between VEA and VIP values [η = (VIP–VEA)/2] for the neutral clusters [48], [49]. It is plotted against n in Figure 4. One can notice the odd-even oscillations pattern for neutral Al_nSb clusters. Apparently, a similar pattern is also observed for Al_n clusters in the range of 10 < n < 16, but not found for n < 10. It further verifies that Al_nSb with odd n and Al_n(n = 2, 3, 5, 6, 7, 8, 9, 12) are more stable than their neighbouring clusters.

Figure 4:

The chemical hardness of the most stable structures vs. cluster size n for the neutral Al_nSb and pure Al_n+1(n = 1–16) clusters.

The thermal stability of clusters can be examined by the DE, the minimum DE corresponds to the favourable dissociation channel (DC). For the neutral and anionic pure Al_n+1(n = 1–16) clusters, the DC Al_n+1 ⟶ Al_n+1−m + Al_m and Al_n+1⁻ ⟶ Al⁻_n+1−m + Al_m are taken into account. The corresponding DE can be estimated by employing the following formulas:

Similarly, the DC Al_nSb ⟶ Al_n−m + Al_mSb for neutral Al_nSb(n = 1–16) clusters, Au_nSb⁻ ⟶ Al_n−m + Al_mSb⁻ (I) and Al_nSb⁻ ⟶ Al⁻_n−m + Au_mSb (II) for anionic Al_nSb(n = 1–16) clusters are employed. The corresponding DE can be calculated as follows:

For the neutral Al_nSb(n = 1–16) clusters:

For the anionic Al_nSb(n = 1–16) clusters:

where E(Al_n+1−m^q), E(Al_m), E(Al_n+1^q), E(Al_n−m), E(Al⁻_n−m), E(Al_mSb), E(Al_mSb⁻) and E(Al_nSb⁻) represent the energy of the corresponding ground state clusters. The most possible DC, the corresponding DEs of the neutral and anionic pure Al_n+1(n = 1–16) cluster and Al_nSb clusters are summarised in Tables S7 and S8 (Supporting Material), and the DEs are plotted against n in Figure 5. Tables S7 and S8 indicate that whether neutral Al_n+1, Al_nSb clusters or anionic Al_n+1 clusters, the DC always tends to dissociate a neutral Al atom. For Al_nSb⁻ clusters, most DCs also prefer to dissociate a neutral Al atom except for AlSb⁻ and Al₁₃Sb⁻ clusters, the DC of AlSb⁻ and Al₁₃Sb⁻ dissociate a neutral Sb atom. Moreover, Tables S7 and S8 and Figure 5 show that Al_n(n = 7, 13), Al₁₃⁻, Al_nSb(n = 3, 5, 12, 14) and Al_nSb⁻(n = 5, 11, 14) clusters possess the larger DE, implying that those clusters have higher stability than other clusters.

Figure 5:

The dissociation energy of the most stable structures vs. cluster size n for the neutral and anionic Al_nSb and pure Al_n+1(n = 1–16) clusters.

Comprehensively considering those sensitive indicators, the relatively more stable structures Al₇Al₁₃⁻, Al₃Sb, Al₅Sb Al₅Sb⁻ and Al₁₄Sb⁻ clusters are determined, respectively. We also find that the more stable clusters Al₇Al₁₃⁻, Al₃Sb, Al₅Sb, Al₅Sb⁻, Al₁₄Sb⁻ have 21, 40, 14, 20, 21, 48 valence electrons, respectively. It seems that Al₅Sb and Al₁₃⁻ possess higher stability due to the magic numbers of 20 and 40 valence electrons, respectively.

3.4 Magnetic Properties

A natural bond orbital analysis is performed based on the most stable structures of neutral and anionic Al_nSb clusters (n = 1–16). The total magnetic moments, and the contributions of Sb, Al to the total magnetic moments of the neutral and anionic Al_nSb clusters (n = 1–16) reinvestigated as shown in Figure 6 and Table 3. Based on the natural bond orbital analysis, one can see that the nature electrons mainly locate on 3s, 3p states for Al atom and 5s, 5p, 4s states for Sb atom, respectively. For neutral Al_nSb clusters (n = 1–16), we can find that, spin orbitals α and β degenerate when n is an odd number except for AlSb (2.01 μ_B), the total magnetic moments are 0 μ_B. Spin orbitals α and β do not degenerate when n is an even number, the magnetic moments can be obtained from the difference in value of the nature electron configurations between α and β spin orbitals. The total magnetic moments are found to be in the range of 0–2.01 μ_B for Al_nSb

Figure 6:

The total magnetic moments, local magnetic moments of Sb and Al atoms for the neutral and anionic Al_nSb and pure Al_n+1(n = 1–16) clusters.

(n = 1–16) clusters, and vary from 0 to 1.08 μ_B for Al_nSb⁻(n = 1–16) clusters. We can also notice that the total magnetic moment of Al_nSb(n = 1–16) is about 1 μ_B for even n, and 0 μ_B for odd n except for AlSb (2.01 μ_B). The total magnetic moment presents obviously an odd-even alternative behaviour for the neutral and anionic Al_nSb clusters (n = 1–16). In addition, the magnetic moment of the cluster is dominated by the spin magnetc moment, but not by the orbital magnetic moment of an electron. For AlSb cluster, the large magnetic moment 2.01 μ_B can be attributed to its spin multiplicity triplet. In order to further analyse the contributions of Sb and Al to the total magnetic moment of the neutral and anionic Al_nSb(n = 1–16) clusters, the magnetic moment of 5s, 5p spin orbitals of the Sb atom and 3s, 3p spin orbitals of the Al atom of the most stable structures for the neutral and anionic Al_nSb(n = 1–16) clusters are investigated in Figure 6 and Table S9 (Supporting Material). Figure 6 and Table S9 also indicate that the total magnetic moment is mainly derived from the contribution of the 3p state of the Al atom for the neutral and anionic Al_nSb(n = 1–16) clusters, while the contribution of 5s state of the Sb atom to the total magnetic moments may be virtually neglected. Moreover, the natural charge shows that the charge is negative for Sb in Al_nSb(n = 1–10, 16) clusters, implying the charge transfers from Al to Sb, while from Sb to Al for Al_nSb(n = 11–15) clusters. For anionic clusters, one negative charge is assigned to Sb and Al atoms except for Al₂Sb⁻(−1.13e) cluster.

Figure 7:

Calculated molecular orbital energy levels and AdNDP chemical bonding analysis of the most stable structure of Al₅Sb clusters together with the molecular orbital maps of the HOMO where, ON represents occupation number.

3.5 Chemical Bonding

Al₅Sb and Al₁₃⁻ clusters present a large energy gap (2.09 and 2.74 eV) and high stability when compared with their neighbouring clusters. It may be ascribed to their magic numbers of 20 and 40 valence electrons corresponding to a closed electronic shell. These electron configurations are well described by the spherical jellium model. We investigated the one-electron energy levels and molecular orbitals (MOs) of the most stable structures of Al₅Sb and Al₁₃⁻ clusters, as plotted in Figures 7 and S4 (Supporting Material). For Al₅Sb cluster, the HOMO occupies a d-type atomic orbital. Lower occupied levels HOMO–1 to –5, except for HOMO–2, resemble d-type atomic orbitals. HOMO–2 level corresponds to s-type molecular orbital, and so does HOMO–9. HOMO–6 to –8 levels possess p-type character. Therefore, according to the assignment of the MO based on the spherical jellium model,the electronic structure of Al₅Sb is best described as 1S²1P⁶1D⁶2S²1D⁴. For the closed shell clusters Al₁₃⁻, the HOMO are fourfold degenerate and labelled as 1F orbitals. HOMO–1 is threefold degenerate (HOMO–1₁, HOMO–1₂ and HOMO–1₃) and labelled as 2P orbitals, where subscripts 1, 2 and 3 serve as a mark to distinguish the degenerate orbitals. 2S and 1S orbitals are occupied as HOMO–3 and HOMO–6 orbitals. Analogously, HOMO–2, HOMO–4 and HOMO–5 are labelled as 1F, 1D and 1P orbitals, which is threefold degenerate, fivefold degenerate and threefold degenerate, respectively. The HOMO degeneracy may be attributed to the high symmetry I_h of the Al₁₃₋ cluster. To further understand the bonding nature, a chemical bonding analysis based on the AdNDP method is performed as shown in Figures 7 and S4. For the Al₅Sb cluster, one lone pairs (LPs), one localised bonds and eight delocalised bonds are found, and the occupation numbers (ONs) vary from 1.87 to 2.00|e|. For one LP with an occupied number ON = 1.87|e|, the main contribution is derived from the Sb but not the Al atom. While the interaction of two peripheral Al atoms plays an important role for 2c-2e σ bond with ON = 1.94|e|. Delocalised bonds are consisted of three 3c-2e σ, two 4c-2e σ, two 5c-2e π and one 6c-2e σ bonds, and the ONs of bonds are very close to the ideal values (2.00|e|). For the Al₁₃⁻ cluster, 21 delocalised bonds with ONs ranging from 1.77 to 1.98 |e| are found. The delocalised bonds consist of 14 3c–2e σ,six 4c-2e σ and one 6c-2e σ bonds. The delocalised 3c-2e σ bonds with ON = 1.88|e| and 1.87|e| are derived from the peripheral Al₃ units, and the 4c-2e σ bonds with ON = 1.77|e| are due to the bonding between the central Al and the peripheral Al₃ units. In addition, the 6c-2e σ bonds with ON = 1.98|e| present a strong interaction among six peripheral Al atoms.