Introduction

Silicon clusters have potential applications in micro-nano devices as building blocks to continue the miniaturization trend of Moore's Law. However, the pure silicon clusters are always chemically reactive, limiting their applications [1, 2]. Fortunately, the introduction of transition metals into silicon clusters can significantly improve their stabilities [3,4,5], and may also introduce novel electronic and magnetic properties [6, 7], possibly allowing them to be used as building block for cluster-assembled materials. For example, the caged clusters of Si16 encapsulating a group-IV metal atom form superatoms with large HOMO–LUMO gaps [8,9,10]. The wheel-like V3Si12− cluster exhibits a ferromagnetic state with a total magnetic moments of 4μB [7]. Therefore, more and more experimental and theoretical efforts have been devoted to this type silicon clusters in recent thirty years [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]. In the early stages, people mainly studied the stability of these clusters through mass spectrometry technology. For example, Beck studied the stability of TMSin (TM = Cr, Mo, W, Cu) clusters by mass spectrometry experiments, revealing several stable clusters including TMSi15,16 (TM = Cr, Mo0/+, W) [11, 12]. Neukermans et al. also studies a series of similar systems including TMSin (TM = Cr, Mn, Cu, Zn) clusters through the similar mass spectrometry experiments [13]. Subsequently, more advanced technology such as photoelectron spectroscopy (PES) was used to investigate the electronic properties of a series of dopants (Sc, Ti, V, Y, Zr, Nb, Lu, Tb, Ho, Hf, Ta, Mo, and W) doped Sin− clusters with sizes n ≤ 20 [14,15,16,17,18]. The demand for understanding the structure information of these type clusters urges the research on their structures. The structures of MSi8-16 (M = Sc–Zn, Zr, Hf, Ta; n ≤ 18) [19,20,21,22,23,24,25,26,27,28], M2Si1-8 (M = Cr, Mn, Fe, Co, Ni) [29, 30], Mo2Si9-16 [31], M2Si11-18 (M2 = Mo2, Nb2, Ta2, W2, NbMo, TaW) [32], Ag/AuSi10,12,14 [51], and AuSi1-120/+ [52] clusters were predicted based on DFT calculations, and their structural evolutions is clarified accordingly. Considering the experimental photoelectron spectrum as the fingerprint of a cluster or molecule [53,54,55], the comparison of experimental and theoretical spectra have been used as criteria to identify the ground-state structures of TMSin− (TM = Sc1-2 [33, 34], V1-2 [35], Cr1-2 [36, 37], Co [38], Nb1-2 [39, 40], Ag [41], La [42], Ta [43], Au1-2 [44, 45], n ≤ 14; TM = Ti1-2 [10], V1-3 [46,47,48], Cr1-2 [47, 49]; 14 ≤ n ≤ 20) clusters in recent years. Both experimental and theoretical results show that TM atoms prefer to be surrounded by Si atoms, and forming endohedral structures with the growing number of Si atoms, except for TM = Cu, Ag and Au for small sizes with TM atoms absorbed on the surface of the bare Sin clusters [41, 44, 50].

Among all 3d transition metals, chromium (3d54s1) is the only one has 6 unpaired electrons in the valence shell, making it an interesting dopant for the modulation of magnetic properties. Therefore, Cr-Si clusters have received widespread attention [36, 37, 47, 49, 56, 57] due to their potential applications in spintronic materials [58] and contact materials [59]. Recent studies have shown that small-sized CrSin exhibits strange structures and magnetic properties [36, 37]. For example, Cr3Si12− possess a wheel-like structure [37]. Both of Cr2Si13− and Cr3Si12− exhibit large magnetic moments of 3μB and 7μB, respectively [37]. Our recent research on CrSi14-18 cluster anions shows that Cr atoms do not always contribute positive magnetic moments [47]. It is worth exploring the structural information and magnetic behaviors of Cr atoms in larger-sized clusters. Based on this, we conducted a systematically theoretical investigation on the structural evolution, electronic and magnetic properties of anionic and neutral CrSin clusters for n = 19–25.

Computational method

The critical step in theoretical study of a cluster is to obtain its reliable structure. Therefore, low-lying structures of CrSin−/0 (n = 19–25) clusters were globally searched through a self-developed genetic algorithm code incorporated with the ORCA 5.0.4 software [60, 61] for DFT calculations. For each size, more than 3000 configurations were generated by the genetic algorithm code to fully search on the potential energy surface. The validity and efficiency of our global optimization algorithm code have been well demonstrated by several of our recent works about TMSi14-20− (TM = V1-3, Cr1-2) [46,47,48,49], TMGe8-17− (TM = Ti-Ta) [62], and In3-16X0,1− (X = Si, Ge) [63], Cr2Ge15-20− [64], CsSi5-16− [65], and Ge4-30− [66] clusters. Considering that the scheme of BP86 functional with Karlsruhe-type basis sets has proven to be suitable for the description of structural evolution and electronic properties of Cr1-2Si14-20− and Cr2Ge15-20− clusters in our previous works [47, 49, 64], therefore, this scheme was also adopted for the calculations of CrSin−/0 (n = 19–25) clusters. Briefly, the def2-SVP basis set [67, 68] and the BP86 functional [69, 70] were adopted for DFT calculations during the global search. Then the higher-quality def2-TZVP basis set [67, 68] was employed to further optimize the top 10–20 candidate isomers to get more accurate geometric structures, and the diffuse def2-TZVP basis set (def2-TZVPD) [71] was adopted to obtain more accurate energies. All the structures have positive vibrational frequencies from vibration analyses. Zero-point-energy corrections were considered for the energy calculations. Various spin multiplicities (SM) were considered in order to obtain the lowest-energy spin state. The vertical detachment energy (VDE) was obtained from the energy differences between the anionic and neutral clusters at the relaxed structure of the anionic state. The adiabatic detachment energy (ADE) were calculated as the energy differences between the anionic cluster and the relaxed neutral clusters using the anionic structure as initial configuration.

The stabilities of anionic and neutral CrSin cluster are evaluated by the average binding energies (Eb) defined as Eqs. (1) and (2) [72, 73]:

$${E}_{b}({{\mathrm{CrSi}}_{n}}^{-})=[\left(n-1\right)E\left(\mathrm{Si}\right)+E\left({\mathrm{Si}}^{-}\right)+E\left(\mathrm{Cr}\right)-E({{\mathrm{CrSi}}_{n}}^{-})]/(n+1)$$
(1)
$${E}_{b}({\mathrm{CrSi}}_{n})=[nE\left(\mathrm{Si}\right)+E\left(\mathrm{Cr}\right)-E({\mathrm{CrSi}}_{n})]/(n+1)$$
(2)

where E(CrSin−) and E(CrSin)is the energies of the anionic and neutral CrSin clusters, respectively; E(Si), E(Si−) and E(Cr) are the total energies of the neutral Si, anionic Si and neutral Cr atoms, respectively. The second order of energy difference (Δ2E) of a CrSin−/0 was calculated using the formula (3):

$${\Delta }_{2}E({{\mathrm{CrSi}}_{n}}^{-/0}) = E({{\mathrm{CrSi}}_{n}}^{-/0}) + E({{\mathrm{CrSi}}_{n}}^{-/0})-2E({{\mathrm{CrSi}}_{n}}^{-/0})$$
(3)

All the graphs of the structures of clusters involved in this work were rendered by using the molecular graphics program VMD 1.9.3 [74]. The average bond length, Wiberg bond order, and electron spin density were calculated by using the post processing program of Multiwfn 3.8 (dev) [75].

Computational results

Structures of CrSin −/0 (n = 19 – 25) clusters

The optimized structures of first three lowest-lying isomers of anionic and neutral CrSi19-25 at the BP86/def2-TZVP level are displayed in Fig. 1. The calculations show that there are four typical endohedral structures (see Fig. 2) as the motifs in these low-lying isomers of CrSi19-25−/0 clusters. These motifs are also adopted for TMSi14 (TM = Mn, Fe, Co) [76], TiSi17-20− [10], and VSi16-20− [46], and CrSi16-18− [49] clusters.

Fig. 1
figure 1

Geometric structures of low-energy isomers for CrSi19-25−/0 clusters. For each size, the total energy difference (eV) with respect to the lowest-lying isomer is provided below the structure, and the symmetry is given inside the parentheses. Gray balls represent chromium atoms. The silicon atoms formed a cage and adsorbed on the surface are highlighted as yellow and red, respectively

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Fig. 2
figure 2

Four typical structural motifs present in isomers of CrSin clusters with n = 19–25 taken from isomers 19A, 22A, 19B, and 20C, respectively. Motifs II and III can be generated from motifs I and IV by removing one of the top Si atoms, respectively

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For anionic CrSin− clusters, the lowest-energy isomers (nAs) prefer to adopt endohedral structures (motif I (Cr@Si14) for sizes n = 19–21, and motif IV (Cr@Si13) for larger ones) as the structural motif with the extra Si atoms attached on the surface. The lowest-energy structures of CrSi19− and CrSi20− also are adopted for VSi19− and VSi20− clusters, respectively [46]. Motif I is adopted for isomers 21B, 21C, and 22B as the core structural motif with excess Si atoms attached to the surface, showing the strong dominance in size range of n = 19–21. Starting from size 22, the dominance of motif I begins to weaken, instead, motif IV begins to show high competitiveness, which is adopted for all the first three isomers for sizes n = 23–25. The motif of isomer 19B is also an endohedral Cr@Si13 structure (motif II), which is also adopted for TMSi14 (TM = Mn, Fe, Co) clusters [76]. The motif III always represents a building block (Cr@Si14) for the ground-state TMSin (TM = Ti−, V−and Cr− for n = 17, 18 [10, 44, 49]) clusters, but is only adopted by isomers 19C, 20B and 20C among all these isomers for n = 19–21, indicating its weak competitiveness in large size clusters.

For neutral CrSin (n = 19–25) clusters, their structures are very similar to that of anionic states. All the lowest-energy isomers of neutral CrSin (n = 19–25) share the same configurations as that of anionic state. The calculated results (see Table S3 of Supplementary materials) show that all the root mean square deviations (RMSD) of anionic and neutral states are small with less than 0.13 Å, except for sizes 20 and 22 with values of 0.221 Å and 0.215 Å, respectively. This result indicates that the reduction of one electron in an anionic system has little effect on its geometric structure. Isomers 19II and 19III of neutral CrSi19 adopt the same configurations as 19B and 19C of anionic state, respectively. Isomers 20II, 20III, 21II, 22III, 23II, 24III, and 25II, adopt the same configurations as 20C, 20B, 21C, 22C, 23C, 24B, and 25B, respectively. Isomers 22II and 23III adopt the motif I, while 24II and 25III are based on the motif IV with excess Si atoms absorbed on the surface.

For sizes n = 21–25 for CrSin−/0 clusters, each successive cluster size is composed of its predecessor with an extra Si atom adsorbed onto the surface, giving evidence of a stepwise growth. Moreover, the core also increases with the cluster size (see Fig. 3). Single Cr atom as core first observed in CrSi9 still exists in CrSi19, but for larger sizes, the clusters prefer to grow by core expansion. Both of the global minimum CrSi20 and CrSi21 possess a two-atom core, while CrSi22 has a three-atom core. The four-atom core of CrSi23 is a tetrahedral bipyramid and is persistent in the global minima of sizes n = 23–25. Significantly, the structure of CrSi23 is very similar to the smallest core–shell silicon cluster Si27− [77], indicating that Cr atom accelerate the formation of cage-like structure for pure silicon cluster. From another point of view, the influence of Cr atom on the structure of pure silicon cluster starts to weaken.

Fig. 3
figure 3

A plot showing the core evolution in the lowest-energy isomers of CrSin (n = 3–25) clusters. Gray and yellow balls represent chromium and silicon atoms, respectively. The results for sizes n = 3–18 are from previous works [36, 49, 78]

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Photoelectron spectra of CrSin − (n = 19 – 25) clusters

Figure 4 shows the simulated PES for the lowest-energy isomers of CrSin− (n = 19–25) cluster anions. For CrSi19−, the spectrum possesses six distinguishable peaks at 3.29, 3.65, 3.79, 4.11, 4.44 and 4.69 eV. In the PES of CrSi20−, there are two small peaks at 3.31 and 3.57 eV, followed by four prominent peaks located at 3.87, 4.26, 4.46, and 4.79 eV. For CrSi21−, the first peak locates at 3.29 eV, followed by three high-intensity peaks centered at 3.89, 4.13, and 4.59 eV. Similar spectrum is also observed in CrSi22−, the PES also possesses a clear small peak at 3.48 eV, followed by three prominent peaks located at 3.85, 4.21, and 4.49 eV. In the case of CrSi23−, there is a large separation between the highest occupied state and (2.92 eV) the next lower-lying state (3.87 eV), indicating a large gap between the highest occupied state (HOMO) and the lowest unoccupied state (LUMO) of the corresponding neutral cluster. This result corresponds to the large HOMO–LUMO gap of 1.50 eV for neutral CrSi23. Similar to CrSi23−, the simulated spectrum of CrSi24− also has a small peak at 3.10 eV and three prominent peaks situated at 3.73, 4.13, and 4.45 eV. For CrSi25−, four discrete peaks are located at 3.12, 3.41, 3.77, and 4.30 eV, respectively. In short, for the simulated PES of CrSin− (n = 19–25) clusters, the spectral characteristics and peak positions are obviously different, indicating their significant differences in electronic properties. The simulated PESs are expected to provide some guidance for future PES measurements.

Fig. 4
figure 4

Theoretical photoelectron spectra of the lowest-lying isomers of CrSin− (n = 19–25). Gaussian broadening with a width of 0.06 eV is used

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Bonding and electronic properties of CrSin −/0 (n = 19 – 25) clusters

To further explore the bonding and electronic properties of the CrSin−/0 (n = 19–25) clusters, the average bond lengths, Wiberg bond orders, vertical detachment energies (VDEs), adiabatic detachment energies (ADE), HOMO–LUMO gaps, average binding energies (Eb), and the second order of energy difference (Δ2E) for the lowest-energy structures were calculated and presented in Figs. 5 and 6 as well as summarized in Table 1.

Fig. 5
figure 5

Average bond lengths (a) and Wiberg bond orders (b) of Si–Si and Si-Cr of the lowest-energy structures of CrSin−/0 (n = 19–25) clusters

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Fig. 6
figure 6

Size-dependent vertical detachment energies (VDE), adiabatic detachment energies (ADE), HOMO − LUMO gaps, binding energies (Eb), and second order of energy difference (Δ2E) of the lowest-lying structures of CrSi19-25−/0 clusters with the results of Si19-25−/0 for comparison. The ground-state structures of Si19-25 clusters are from previous works [77, 79]

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Table 1 Spin multiplicities (SM), average bond lengths (BL, in Å), bond orders (BO), vertical detachment energies (VDE, in eV), adiabatic detachment energies (ADE, in eV), HOMO − LUMO gaps (EHL, in eV), average binding energies (Eb, in eV), and the second order of energy difference (Δ2E, in eV) of the lowest-energy structures of CrSin−/0 (n = 19–25) clusters
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From Fig. 5, the average bond length and bond order of Si–Si and Si-Cr bonds in anionic state are close to those in the neutral state. From Fig. 5(a), the average Si-Cr bond lengths (2.649 Å–2.819 Å) have significantly larger values than that of Si–Si bonds in the range of 2.446 Å–2.545 Å. The Wiberg bond orders (Fig. 5b) of Si-Cr bonds show a gradual decrease trend for CrSin−/0 (n = 19–25) clusters in range of 0.495–0.376, while that of Si–Si bond fluctuate in a small size range of 0.374–0.405.

It shows that the VDEs (Fig. 6a) of CrSin− are very close for n = 19 to 21, rise to 3.48 eV at n = 22, and then abruptly drop to 2.92 eV at n = 23, gradually rising up again from n = 24 to 25. The trend of ADEs of CrSin is roughly consistent with that of VDE, except for sizes 20 and 22, which corresponds to the large root mean square deviations (RMSD, see Fig. S3) values of anionic and neutral structures. The trend of VDE of Sin− is roughly consistent with that of ADE for Sin. All the ADE and VDE values for neutral and anionic Sin are bigger than that of CrSin clusters, except for sizes 19 and 22, respectively. This result indicates that Cr atoms can usually reduce the detachment energy of silicon cluster.

The trend of the HOMO–LUMO gaps for CrSin− (in Fig. 6b) is similar to that of the VDE, except for size 23. The HOMO–LUMO gap of neutral CrSin and Sin clusters are significantly larger than that of anionic states. This is because there are unpaired electrons in the anionic states. It is worth noting that the neutral CrSi22 and CrSi23 have large HOMO–LUMO gaps of 1.37 eV and 1.50 eV, respectively. All the HOMO–LUMO gap values for neutral and anionic CrSin are bigger than that of Sin clusters, except for sizes 19 and 25. It is worth noting that the influence of Cr atoms on the HOMO–LUMO gap value of Sin− is generally very small, but for the neutral state.

From Fig. 6c, the average binding energies (Eb) exhibit similar trends in both anionic and neutral CrSin and Sin clusters. Eb of CrSin− for sizes 19 and 20 are significantly lower than that of larger sizes, increase gradually until coming to the maximum at n = 22, and then decrease monotonously. For neutral CrSin clusters, the largest value of Eb occurs in size 23. Compared with pure Sin clusters, CrSin clusters have a larger average binding energy, which is consistent with our expected results that doping transition metals can improve the stability of pure silicon clusters.

Interestingly, the trends of the Δ2E (Fig. 6d) of CrSin− and Sin− clusters are consistent with that of average binding energy curves. CrSi22− and CrSi22 also possess the largest Δ2E values, indicating the largest relative stabilities. For anionic and neutral Si clusters, their local maximum values occur at sizes 21 and 22, respectively. Significantly, the CrSi22 and CrSi23 clusters have the considerable HOMO–LUMO gap, Eb, and Δ2E values in the neutral and anionic states, respectively. The results indicate that these two clusters have high stabilities among these clusters.

Magnetic properties of CrSin − (n = 19 – 25) clusters

Our results (see Tables S1 and S2 in Supplementary materials) show that all these of anionic and neutral CrSi19-25 clusters are in their lowest spin states, therefore, we just explore the magnetic properties of the CrSi19-25− cluster anions by the calculations of spin electron density and Hirshfeld population. The results are displayed in Fig. 7 and listed in Table 2, respectively. From Fig. 7, one can see that the green zones (the excess alpha electrons) appear around the Cr atom and some Si atoms with low coordination number, while the red regions (the excess beta electrons) are hardly distributed. Different from the results that Cr atoms in small-sized CrSin− (n ≤ 18) clusters [49] contribute most of the total magnetic moments of 1 μB, their contributions in larger clusters are not significant, even less than that of some Si atoms, except for size 22 with μCr = 0.55 μB. This result indicates that the source of magnetic moment changes obviously for larger sized clusters studied here, which may be related to the saturation of Cr atoms.

Fig. 7
figure 7

Isosurface maps of the electron spin density of the lowest-energy CrSi19-25− clusters. The isosurface is set to ± 0.006 e/Å3. Green and red isosurfaces indicate the positive and negative electron spin density, respectively

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Table 2 The total magnetic moments (μT) of the CrSi19-25− clusters were obtained by Hirshfeld population analysis, along with the local magnetic moment on the Cr atom (μCr) and the Si atom with the maximum value (μSi-Max). All magnetic moments are in μB
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Conclusions

The structural evolution, electronic and magnetic properties of CrSi19-25−/0 clusters have been computationally investigated. Global research for of the minimum structures of these clusters have been performed on a self-developed genetic algorithm code combined with DFT calculations. All the anionic CrSin− clusters share the same configurations as that of their neutral states for n = 19–25. All these CrSi19-25−/0 clusters prefer to adopt endohedral structures (Cr@Si14 for sizes n = 19–21, and Cr@Si13 for n = 22–25) as the structural motif with the remaining Si atoms attached on the surface. The simulated photoelectron spectra show that these clusters have obvious size dependence. From the average binding energy results, the doping Cr atoms can significantly improve the stabilities of the pure silicon clusters. Among all these clusters, both of CrSi22 and CrSi23 possess large HOMO − LUMO gaps, average binding energies, and the second order of energy differences, indicating their high stabilities. All of the clusters possess total magnetic moment of 1 μB, but with very different contributions from that of small sizes (n ≤ 18), except for size 22.