Geometrical and electronic properties of PdWSi_n (n=10–20) semiconductor materials

Computational details

In order to provide the accurate and reliable geometrical and electronic properties of the hetero atomic bimetal Pd- and W-doped silicon clusters at the size range (10≤n≤20), the hetero atomic W- and Pd-doped silicon clusters are calculated at the unrestricted B3LYP/LanL2DZ level with relativistic effect taken into account (Han et al., 2004, 2007; Zhao et al., 2014a). The standard LanL2DZ basis sets are employed to provide an effective way to reduce difficulties in calculations of two-electron integrals caused by heavy metals W and Pd atoms. All computational works were carried out using the Gaussian 09 package (Frisch et al., 2009). The polarization basis sets are not considered in this work because our previous work showed that they are insignificant as the calculated results with and without polarization basis sets provide similar tendencies of the calculated electronic properties. Based on prior studies of the single TM-doped Si_n clusters (Wang and Han, 2005; Han and Hagelberg, 2009; Fan et al., 2010; Zhao et al., 2014a, 2015, 2016a, 2017a), only a small number of structures were tried for each size n. For the large-size clusters containing 8–18 atoms (Han et al., 2004), there are a large number of local minima for each cluster, whose global minimum is difficult to obtain simply according to the calculated total energies of the isomers. When the unstable geometry with imaginary frequencies is found, a relaxation along the coordinates of the imaginary vibrational mode is rearranged until a true local minimum is finally reached. In addition, the stability is examined by calculating the harmonic vibrational frequencies. Therefore, geometries and total energies for each stable cluster and its stable isomers actually correspond to the local minima.

In order to obtain possible structures of the most stable Pd- and W-doped Si_n (n=10–20) clusters, some previous available theoretical results on single TM-doped silicon clusters (Re, Cu, and Zr) are considered as our initial geometries (Han et al., 2004, 2007; Zhao et al., 2014a). Then, the equilibrium geometry is determined by adding another TM on the geometry starting from high-symmetry structure to low-symmetry structure. This leads to a limited number of possible stable structures for each size of cluster.

The respective Si₂, WPd, WSi, and PdSi molecules are used to test the reliability of our calculations. Their calculated bond lengths, vibrational frequencies, and dissociation energies are illustrated in Table 1. As can be seen from Table 1, one finds that our calculated results of Si₂, PdW, WSi, and PdSi clusters are in good agreement with the reported theoretical and experimental data (Huber and Herzberg, 1979). The calculated dissociation energy of the PdSi cluster is smaller than that of Si₂ cluster but bigger than that of the PdW molecule; however, the dissociation energy of WSi is bigger than those of respective Si₂ and PdSi. These findings indicate that in PdWSi_n clusters the interactions of Si-Si and W-Si are stronger than those of Pd-Si and Pd-W, and W-Si interaction plays a dominant role in stabilizing the PdWSi_n clusters.

Geometries and stabilities

The stable geometries of PdWSi₁₀ isomers are obtained (Figure 1 ): 10a is optimized as the most stable isomer, 10a and 10b are described as single Pd surface capped on deformed D_5h silicon frame with W being localized at the center of the silicon frame while 10c and 10e are described as W surface capped on deformed D_5h geometry with Pd being localized at the center of the silicon frame. Surprisingly, 10d and 10g can be seen as Pd and W atoms being surface capped on deviated D_5h Si₁₀ atoms; it is lower in stability than 10a. In general, the dominant forms of PdWSi₁₀ geometries are D_5h pentagonal prism, and W center localized at PdWSi₁₀ isomers are more stable than Pd center localized geometries because W-Si interactions are the strongest one (Table 1).

Figure 1:

The stable PdWSi_n geometries.

11a is optimized as the most stable geometry while 11b is a slightly deformed D_6h hexagonal prism with one Si atom being replaced by a Pd atom. On the basis of the optimized geometries of Pd- and W-doped Si₁₁ clusters, it is obvious that clusters with W being doped into the silicon frame and Pd atom being surface capped silicon frame are more stable than the clusters with Pd being doped into the silicon frame and W atom being surface capped into the silicon frame. In addition, the Pd-W bond lengths in all stable geometries are distinctly elongated under the interactions of Pd-Si and W-Si because Pd-Si and W-Si interactions in PdWSi_n are stronger than those of Pd-W interactions. Thus, W center clusters are favorable geometries for PdWSi₁₁ clusters in that W-Si interactions are stronger than those of Si-Si interactions.

12a is yielded after one Pd atom is vertically surface capped on a D_6h WSi₁₂ unit, which is the most stable geometry. 12b can be seen as paralleled Pd being surface capped on a slightly deformed D_6h hexagonal prism. However, the 12c and 12d with surface localized W atom are irregular geometries; their energies are higher than those of W center localized silicon clusters.

As for the PdWSi₁₃ isomers, seven stable isomers are presented and discussed. 13a can be seen as one Si atom being surface capped on a slightly deformed D_6h PdWSi₁₂ unit, which is the most stable geometry. 13b is obtained by adding two Si atoms on the surface of 11a while 13c can be seen as one Si and Pd atom being surface capped on the respective top and bottom of W center localized D_4d Si₁₂ frame. 13a, 13b, and 13c with W atom being centered into the silicon frame are lower in energies than the 13d–13g isomers. Seriously deformed 13d–13f are generated with both Pd and W atoms being sinked into silicon frame with quadrangular prism forms; these geometries are higher in energies than the former (13a–13c). 13g with Pd completely encapsulated into the silicon frame is the highest energy isomer (Table 2 and Figure 1), and it is obvious that one W atom is capped on the surface of open cage-like PdWSi₁₃ frame.

Guided from PdWSi₁₃ isomers above, seven stable PdWSi₁₄ isomers are discussed. 14a is generated after one Si atom is being capped on an arranged 13c isomer while 14b and 14c are generated after two Si atoms are being capped on the respective 12b and 12d. The optimized geometries show that W center localized silicon frames with Pd atom being capped on the WSi₁₄ surface are lower in total energies. According to the optimized geometry of 14a, it is obvious that 14a is enhanced in stability than Pd-doped 14g and 14f geometries. Furthermore, 14g and 14f can be seen as Pd and W being completely doped into the deformed heptagonal Si₁₄ prism, and 14f has the smallest dipole moment of 0.13 Debye. In analogous to the geometries of PdWSi₁₃ isomers, the dominant geometry of PdWSi₁₄ isomers is still W center doped and Pd surface capped open cage-like geometry.

15a, which is obtained by surface capping one silicon atom on 14a, is the most stable isomer. This geometry deforms D_5d symmetry due to the replacement of one silicon atom by one Pd atom. 15b, 15c, and 15e are the derivatives of the 15a isomer, which are higher in energies than the 15a isomer. 15d, 15f, and 15h are obtained after one silicon atom is surface capped on 14f and 14g isomers. Apparently, the geometry with the heptagonal prism is lower in energy. Particularly, Pd and W atoms being completely encapsulated into Si₁₅ frame in all stable PdWSi₁₅ geometries become dominant geometries; consequently, cage-like geometries are formed. Furthermore, Pd and W atoms in the cage-like geometries interact with more silicon atoms simultaneously with unequivalent Pd-Si or W-Si bond lengths and try to saturate the dangling bonds of silicon atoms with delocalized covalent bonds.

Bimetal-doped deformed C_2v16a is optimized as the most stable PdWSi₁₆ cluster, which is composed of two slightly deformed D_5h WSi₁₀ and PdSi₁₀ units. Consequently, the PdWSi_n cluster with n=16 is the critical size for Pd and W atoms being completely encapsulated into Si₁₆ frame. Interestingly, this geometry is similar to those of Pd₂Si₁₆ and Pt₂Si₁₆ (Han et al., 2007; Zhao et al., 2014a). 16b and 16c can be seen as an arrangement of 16a geometry; 16b is composed of a D_5h WSi₁₀ unit and a D_4h PdSi₈ unit, and it is lower in total energy than 16a, while 16d, which can be seen as paralleled Pd and W being doped on deformed D_8h Si₁₆ frame, is lower in stability than 16a and 16b; reflecting that geometrical arrangement can alter total energy and stability, 16d is an unfavorable geometry. 16f is an irregular geometry with Pd atom being a surface-capped silicon frame.

Five stable WPdSi₁₇ and five stable PdWSi₁₈ as well as PdWSi₁₉ geometries are obtained by adding silicon atoms on the corresponding PdWSi₁₆ frame. According to the calculated results, it is obvious that PdWSi₁₇ and PdWSi₁₈ as well as PdWSi₁₉ geometries are generally similar in frame to those of PdWSi₁₆ geometries with TM atoms being completely encapsulated into the Si frame. The tendency of forming a closed cage-like geometry with Pd and W being completely encapsulated into the silicon frame is apparently exhibited. Additionally, 18b with smaller dipole moment has a higher symmetrical geometry.

PdWSi₂₀ geometries are optimized as the stable geometries, and fullerene-like PdWSi₂₀ geometries are the most favorable structure. 20a, which can be seen as two Si atoms being capped 18a, is the most stable geometry (Figure 1 and Table 2). This study will provide a new way for investigation on hetero TM atoms doped silicon clusters. Especially, cage-like semiconductor materials are interesting and cause a sensation (Zhao et al., 2018).

Charge-transfer mechanism

The calculated Mullikan and Natural atomic populations for the most stable PdWSi_n (n=10–20) are listed in Table 3 . It is apparent that electronic charges are transferred from the silicon frame to W and/or Pd (larger sized cluster) atoms; in other words, W (Pd at large size) in the most stable PdWSi_n (n=10–20) clusters obtains electronic charges from surrounding Si atoms. Furthermore, the encapsulated Pd and W atoms in Si frame obtain more electronic changes than Pd atom being surface capped on the Si surface because the encapsulated TM atom tends to interact with more silicon atoms with unequivalent bond lengths and to saturate the dangling bonds of silicon atoms. On the basis of these studies, internal electron transfer within PdWSi_n clusters in their equilibrium ground structures proceeds from the Si frame to Pd and W atoms; i.e. the TMs behave as the electron acceptors. This pattern seems to be rooted in the tendency of Pd and W atoms to attain a completely filled 5d¹⁰6s² configuration. Therefore, the doped Pd and W atoms can stable PdWSi_n (n=10–20) geometries. In addition, Mullikan populations are not accurate and mislead the charge-transfer in TM containing silicon systems (Han and Hagelberg, 2001a; Han and Shi, 2001; Han et al., 2004).

Relative stabilities

The averaged atomic binding energies [E_b(n)] of WPdSi_n clusters and the fragmentation energies [D(n, n−1)] for WPdSi_n clusters with respect to removal of one Si atom from the most stable PdWSi_n clusters are calculated and discussed. E_b(n) and D(n, n−1) of PdWSi_n (n=10–20) clusters are defined as follows:

D(n, n−1)=ET(PdWSin−1)+ET(Si)−ET(PdWSin)

Eb(n)=ET(Pd)+ET(W)+nET(Si)−ET(PdWSin)n+2

where E_T(Si), E_T(Pd), E_T(W), and E_T(PdWSi_n) represent the total energies of the respective Si, Pd, W, and PdWSi_n clusters. The calculated E_b(n) and D(n, n−1) values of the most stable PdWSi_n (n=10–20) isomers are plotted as the curves of E_b and D(n, n−1) against the corresponding number of the Si atoms in Figure 2 and Table 4. The features of size evolution are intuitively observed, and the peaks of the curves correspond to those clusters that have enhanced local stabilities in comparison with their neighbors.

Figure 2:

The calculated averaged atomic binding energies and fragmentation energies.

The calculated E_b(n) and D(n, n−1) of PdWSi_n clusters are shown in Figure 2 and Table 4. The curve of E_b(n) in Figure 2 exhibits that the averaged atomic binding energies are increased as the size of Si atoms are being increased. According to the calculated fragmentation energies shown in Figure 2, the remarkable peak at n=14 for W- and Pd-doped Si_n (n=10–20) clusters is exhibited, reflecting that the corresponding cluster has the strongest relative stability and large abundance in mass spectroscopy as compared with their corresponding neighbors. As can be seen from the curves shown in Figure 2, the particularly most stable geometry can be assigned to the PdWSi₁₄ cluster, which has higher fragmentation energy and stability than the other PdWSi_n. Interestingly, these findings of Pd- and W-doped silicon clusters are different from those of the single TM-doped silicon clusters (Wang and Han, 2005). Furthermore, the calculated fragmentation energy in terms of the 14a isomer shows that it has the strongest stability of all the PdWSi_n isomers, reflecting that the 14a structure is the preferred geometry. In addition, our predicted critical size for Pd and W completely encapsulated into the silicon frame is the same as that of Pd₂-encapsulated silicon frame.

HOMO-LUMO gaps

The electronic properties of PdWSi_n clusters are discussed by examining the energy gaps between HOMO and LUMO levels. The calculated HOMO-LUMO gaps are tabulated in Table 1 and plotted in Figure 3; three peaks with n=12, 16, and 19 are apparently shown in the curve of Figure 3, indicating that PdWSi₁₂, PdWSi₁₆, and PdWSi₁₉ clusters have larger HOMO-LUMO gaps, which are actually weakened in chemical activities than their neighboring clusters. As can be seen from Figure 3, HOMO-LUMO gaps are generally oscillatory decreased as n goes from 12 to 20. Additionally, the PdWSi₁₂ cluster has the biggest HOMO-LUMO gap with stable chemical activity; furthermore, this cluster can be used as building blocks for new semiconductor materials.

Figure 3:

The calculated HOMO-LUMO gaps for the most stable PdWSi_n (n=10–20) clusters.