Geometrical stabilities and electronic properties of Si_n (n = 12–20) clusters with rare earth holmium impurity: a density functional investigation

Abstract

HoSi_n (n = 12–20) clusters with different spin states have been systematically investigated by using density functional theory with the generalized gradient approximation. The total energies, growth-pattern and equilibrium geometries as well as the APT charges of the HoSi_n (n = 12–20) clusters are calculated. The relative stabilities in terms of the calculated atomic averaged binding energies and fragmentation energies are discussed, revealing that the cake-like HoSi_n (n = 16, 18, 20) clusters have enhanced stabilities. Furthermore, the highest occupied molecular orbital – lowest unoccupied molecular orbital (HOMO–LUMO) gaps of the HoSi_n (n = 12–17) are above 1.55 eV while HoSi₁₆ has the largest one (1.95 eV). Interestingly, the calculated dipole moments of the cake-like HoSi_n (n = 16, 18, 20) clusters are very small, corresponding to the global minima. According to the calculated APT charges of the Ho atom in the HoSi_n (n = 12–20) clusters, the contribution of charge-transfer to the stability of HoSi_n clusters is briefly discussed, manifesting that the charges in HoSi_n clusters transfer from the Si atoms to the Ho atom. Additionally, the optimized geometries show that the rare earth Ho atom is completely encapsulated into the centre of the Si frame at n = 15. This finding is in good agreement with the available experimental results.

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1. Introduction

Silicon and germanium clusters have been studied extensively by using both theoretical and experimental techniques, as they may be employed not only as model systems for investigating localized effects in the condensed phase, but also as building blocks for developing new silicon based nanomaterials with tunable properties.^1–18 However, pure silicon clusters are unsuitable as building blocks because they are chemically reactive due to the existence of dangling bonds. Transition metal atom-doped silicon clusters, on the other hand, may tend to form closed-shell electronic structures that are of higher stability than the pure silicon clusters and represent a new class of endohedral clusters encapsulating metal atoms^3–18 and have important applications in the semiconductor industry. What’s more, the properties can be changed greatly with the aid of the addition of “impurity” atoms.^4,12,15 In addition, the transition metal (TM) atoms can saturate the dangling bonds on the surface of the Si frame, which stabilizes the Si cages. Encouraged by these results, we believe that searching for suitable impurity atoms to obtain silicon-based nanomaterials with the desired properties is possible.

Rare earth lanthanide metal (Ln)-doped silicon clusters have attracted wide interest. This is because the Ln atoms could retain significant portions of their magnetic moments even when enclosed by a silicon cage due to their localized f-electrons. Experimentally, Ohara et al. firstly investigated the TbSi_n⁻ (n = 6–16) clusters by using photoelectron spectra (PES) and a chemical-probe method. The Tb-doped silicon clusters are relatively more stable with regard to photofragmentation than the bare silicon clusters of a similar size.² The electronic properties of silicon clusters containing a transition or lanthanide metal atom, MSi_n (M = Sc, Ti, V, Y, Zr, Nb, Lu, Tb, Ho, Hf, and Ta), were investigated using anion photoelectron spectroscopy at 213 nm. In the case of the elements Sc, Y, Lu, Tb, and Ho, the threshold energy of electron detachment exhibits local maxima at n = 10 and 16. The electronic characteristics of MSi_n are closely related to their geometric and electronic structures.³ Grubisic et al. carried out the study of LnSi_n⁻ (n = 3–13, Ln = Ho, Gd, Pr, Sm, Eu, and Yb) clusters,⁵ and found that the dramatically increased adiabatic electron affinities of the LnSi_n clusters were attributed to their inherent electronic stabilization. Stimulated by these experimental observations, several computational investigations have been performed for Ln-doped silicon clusters including YbSi_n (n = 1–13),^4,8,9 EuSi_n (n = 1–13), and LuSi_n (n = 1–12) clusters.^6–10 The calculated results indicate that the Ln in the lowest-energy LnSi_n geometries occupied a gradual sinking site, and the site varies between the Ln surface-absorbed forms to the Ln-encapsulated forms with the size of the Si_n atoms increasing. Moreover, the charge-transfer between the Ln atom and the Si_n framework depends on different kinds of Ln and the cluster size. The stabilities for the specific-sized Ln-silicon clusters are enhanced after the Ln is doped into the Si_n frame.

In a way, rare earth metals are special transition metals, possessing many of the properties of these elements (such as the optical and magnetic properties, and so on). Although there is some research on the properties of Ln-doped silicon clusters, only systematic theoretical investigations on the electronic properties and geometries of small-sized HoSi_n (n = 1–12, 16, 18)¹⁰ and HoSi_n (n = 1–12, 20) clusters have been reported so far; however, the calculated results can not provide the threshold size of the Ho atom being encapsulated into the silicon frame and do not allow exploration of the growth-patterns and the cake-like Ho-doped Si geometries.

In order to understand the properties of HoSi_n clusters and the critical size of the Ho encapsulated into the Si frame and to reveal the unusual size-dependent properties of the rare earth Ho-doped Si_n clusters, we carry out a detailed study of HoSi_n (n = 12–20) clusters with relativistic effects being taken into account. The main objective of this research, therefore, is to provide a detailed investigation of the equilibrium geometries, charge-transfer properties, relative stabilities, fragmentation energies [D(n,n − 1)], atomic averaged binding energies [E_b(n)], and HOMO–LUMO gaps of HoSi_n (n = 14–20) clusters, which can provide instrumental help in the quest for this type of cluster-assembled material. However, it should be pointed out that the relativistic effect of the rare earth Ln is a challenging problem because of the complicated 4f electrons.

2. Computational details

The explicit treatment of all the electrons in a cluster (including the heavy metal Ho element) having a large number of atoms constitutes a demanding computational task. One of the best ways to surmount this difficulty is to make use of electron core potentials (ECP), also known as relativistic pseudopotentials,¹⁹ by means of which only the valence electrons are explicitly treated. ECP calculations can actually provide accurate results for both homo- and heteronuclear clusters and their various combinations as firmly proved by previous investigations on actinide elements and silicon/germanium clusters.^20,21 Therefore, the combination of density functional theory (DFT) methods with ECP provides a feasible and accurate approach to the electronic structure study of the HoSi_n clusters as shown below.

Present calculations are done at the level of the DFT with the hybrid exchange and correlation (X3LYP) functional in combination with the 6-31G basis sets for Si atoms and the Large-core Stuttgart effective core potentials ECP56MHF ECP for describing the Ho atom,¹⁹ as implemented in the Gaussian 09 code.²² The various HoSi_n isomers, optimized geometries, and their harmonic vibrational frequencies are obtained. With the aim to examine all stable geometries as minima and to estimate zero-point energy corrections that are included into the calculated electronic energies, vibrational frequencies are calculated at the same level of theory.

In order to test the reliability of our calculations, the HoSi molecule is calculated at the X3LYP level with ECP56MHF ECP for the Ho atom and 6-31G for the silicon atom. The calculated bond length, frequency, and electronic state is 2.733 Å, 288.6 cm⁻¹, and ⁴Σ, respectively; this is in good agreement with the previously calculated values of 2.838 and 218.8 cm⁻¹ at the mPW3PBE level in combination with the 6-31G* basis sets for the Si atom and the Large-core Stuttgart effective core potentials (ECP28MWB) ECP for describing the Ho atom,¹⁶ as well as the values (2.773 Å, 267.3 cm⁻¹, and ⁴Σ) calculated at the X3LYP level with ECP56MWB ECP for the Ho atom and the 6-31G basis sets for the silicon atom, as well as those values (2.766 Å, 265 cm⁻¹) calculated at the QCISD level with the same basis sets. The Si–Si bond length, frequency, and the electronic state of Si₂ at the X3LYP level with 6-31G basis sets is calculated to be 2.236 Å, 507.4 cm⁻¹, and ³Σ_g, respectively, which is in good agreement with the experimental results (2.246 Å, 510.98 cm⁻¹, and ³Σ_g)²³ and the calculated values of Si₂ of 2.352 Å, 445.7 cm⁻¹, and ³Σ_g, respectively, at the B3LYP/LanL2DZ level.^17,24,25 The calculated bond length, frequency, and electronic state for the Ho₂ dimer at the X3LYP level with ECP56MHF ECP for the Ho atoms is 3.239 Å, 110.7 cm⁻¹, and ⁵Σ_g, respectively, which matches the calculated bond length of the Ho₂ dimer (3.875 Å),²⁶ however, it is not in agreement with a previous result of 4.727 Å;²⁷ however, the values are in good agreement with the values (3.048 Å, 127.4 cm⁻¹, and ⁵Σ_g) calculated at the X3LYP level with ECP56MWB ECP for the Ho atoms as well as the values (3.078 Å, 131.7 cm⁻¹) at the QCISD level with ECP56MWB ECP basis sets. Consequently, the X3LYP method with ECP56MHF for the Ho atom and 6-31G basis sets for the Si atom is reliable enough to be applied to describe the properties of the HoSi_n clusters.

As far as the Ho-doped silicon clusters are concerned,^5,10 the previously calculated results, without considering relativistic effects, can not match available experimental results and find the critical size of Ho encapsulated within silicon clusters. In order to explore the accurate and reliable critical size of Ho completely encapsulated into silicon clusters as well as the physical and chemical properties of the HoSi_n (n = 12–20) clusters, the systematic investigations on the different sized Ho-doped Si_n clusters are carried out with relativistic effects being taken into account. Equilibrium geometries of the HoSi_n (n = 12–20) clusters and their relative stabilities are investigated, and their frequencies are examined. If the imaginary frequencies are found, a relaxation along the coordinates of the found imaginary vibrational mode will be carried out until the local minimum is actually obtained. In addition, the spin electronic state for each initial geometry is considered at least from spin S = 1/2, 3/2, 5/2, to 7/2, and to 9/2.

3. Results and discussion

3.1. Geometries and stabilities

HoSi₁₂. Different from the transition metal (TM)-doped Si₁₂ (TM = Re, W, Pd, and Zr) clusters,^12,15,24,25 the deformed hexagonal prism, D_6h 12a and 12b isomers, with Ho being capped onto the Si₁₂ structure, is generated and its total energy is much lower than that of the other Ho-doped Si₁₂ structures (Fig. 1). As seen from the examined equilibrium geometries of the HoSi₁₂ clusters, the identified structures for the most stable HoSi₁₂ are usually different from the ReSi₁₂, ZrSi₁₂, and WSi₁₂ clusters. The two cluster series show different growth patterns in that the critical sizes for the TMSi_n (TM = Re, Zr, and W) endohedral geometries are specified to be n = 10 or 12 while the critical size for the Ho atom in the HoSi_n clusters being completely encapsulated into the caged silicon frame turns out to not be n = 12. The 12c and 12d isomers deviate to the tetragonal and pentagonal prisms with the Ho atom being localized at the surface site of the silicon frameworks; their total energies are higher than the most stable 12a isomer (Table 1). The 12e isomer with the Ho atom acting as one of the silicon atoms is the highest one in terms of total energy. One concludes that the geometries at the size of n = 12 are dominated by the Ho surface-capped structures.

Fig. 1 The stable HoSi_n (n = 12–20) geometries.

Table 1 Symmetries (sym), APT, dipole moment (Dip), the HOMO and LUMO gaps (E_gap), the lowest frequencies (Freq), spin (S), total energy (E_T), relative energy (ΔE), and S² operator of HoSi_n (n = 12–20) clusters

System	Isomer	APT	S	S^2	Dip (Debye)	Freq (cm^−1)	Egap (eV)	ET (Hartree)	ΔE (eV)
HoSi12	12a	−0.16	1/2	0.79	0.61	35.4	1.782	−3509.429974	0.00
	12b		1/2	0.77		73.1		−3509.425637	0.12
	12c		1/2	0.767		56.3		−3509.424052	0.16
	12d		1/2	0.77		61.9		−3509.416065	0.38
	12e		1/2	0.79		30.2		−3509.415085	0.41
HoSi13	13a	0.01	1/2	0.78	1.45	73.8	1.866	−3798.873102	0.00
	13b		1/2	0.78		55.6		−3798.854091	0.52
	13c		1/2	0.77		41.5		−3798.850846	0.61
	13d		1/2	0.77		54.1		−3798.850604	0.612
HoSi14	14a	−0.77	1/2	0.79	1.08	50.9	1.81	−4088.305069	0.00
	14b		1/2	0.79		6.3		−4088.301805	0.09
	14c		1/2	0.78		45.8		−4088.295465	0.26
	14d		1/2	0.78		52.5		−4088.288696	0.45
	14e		1/2	0.76		64.2		−4088.301875	0.086
HoSi15	15a	−1.12	1/2	0.78	1.62	39.2	1.67	−4377.763301	0.00
	15b		1/2	0.77		44.2	1.58	−4377.736937	0.72
	15c		1/2	0.78		35.7		−4377.732548	0.84
	15d		1/2	0.76		35.6		−4377.730276	0.90
HoSi16	16a	−1.78	1/2	0.79	0.03	36.3	1.95	−4667.229364	0.00
	16b		1/2	0.77		37.4		−4667.197371	0.87
	16c		1/2	0.78		56.5		−4667.193831	0.97
	16d		1/2	0.77		62.0		−4667.190793	1.05
HoSi17	17a	−1.20	1/2	0.79	0.71	62.0	1.55	−4956.651556	0.00
	17b		1/2	0.78		55.4		−4956.648025	0.096
	17c		1/2	0.77		27.4		−4956.606087	1.237
	17d		1/2	0.76		41.7		−4956.597174	1.48
HoSi18	18a	−1.09	1/2	0.77	0.31	35.3	1.30	−5246.133360	0.00
	18b		1/2	0.77		71.8		−5246.113030	0.553
	18c		1/2	0.78		42.9		−5246.085834	1.293
	18d		1/2	0.76		50.1		−5246.070841	1.701
	18e		1/2	0.77		56.8		−5246.116888	0.45
HoSi19	19a	−1.42	1/2	0.78	1.34	43.3	1.12	−5535.567366	0.00
	19b		1/2	0.78		46.5		−5535.473914	2.54
	19c		1/2	0.76		43.7		−5535.469922	2.65
	19d		1/2	0.77		36.6		−5535.463601	2.82
HoSi20	20a	−0.94	1/2	0.77	0.00	41.6	1.45	−5825.056184	0.00
	20b		1/2	0.78		32.3		−5824.933468	3.34
	20c		1/2	0.77		29.9		−5824.911108	3.95

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HoSi₁₃. The 13a geometry is generated after one silicon atom is added onto the 12c isomer. Its total energy is the lowest one. As seen from Fig. 1, it is obvious that the Ho atom still localizes at the surface site of the silicon frame and interacts with the silicon atoms. Furthermore, 13a forms a cage-like geometry, and the Ho atom in 13a does not tend to be moved into the centre of the silicon frame. On the contrary, the 13b geometry with the Ho atom being localized at the centre of the deformed hexagonal prism is higher in energy than 13a. It is generated from 12d. As for the 13c and 13d isomers, the Ho atom acts as a surface silicon atom and the low-lying geometries are formed.

HoSi₁₄. n = 14 is the critical size for Zr atom-doped silicon clusters. The Zr atom is completely encapsulated into the centre of the silicon frame. As can be seen from the findings related to the geometries of the examined systems of HoSi₁₄, the 14a and 14b isomers are seen with the Ho atom being surface-capped on the irregular and seriously distorted hexagonal prism Si₁₂ frame; these geometries are yielded after a silicon atom is added to the surface of 13b. As for the 14c isomer, it can be described as the Ho atom being encapsulated into the centre site of the distorted heptagonal prism D_7h Si₁₄ frame with the Ho atom interacting with all silicon atoms; its total energy is slightly higher than that of the 14a and 14b isomers. The 14d geometry with a lower stability and higher energy deviates to the 14a and 14b geometries. The Ho atom in the 14d geometry acts as a surface silicon atom and contributes to form a pentagonal prism. Generally, the Ho atom in the 14a geometry shows a tendency to be moved into the silicon frame. On the basis of this finding, one can predict that the Ho atom is squeezed out of the silicon cage when it is small because Ho needs a larger cage in order to be situated within it.

HoSi₁₅. The threshold size of the endohedral HoSi_n clusters turns out to be n = 15. The observed 15a structure, which can be seen as evidence of the Ho-encapsulated silicon framework, is the most stable structure; in comparison with the 14a geometry, the Ho in the 15a isomer, which can be described as the Ho atom interacting with silicon atoms directly, moves gradually into the centre site of two hexagons of the silicon framework; this geometry being generated after a silicon atom is added to the surface in the 14a frame is different from the most stable structures of the TM-doped Si_n (TM = Mo and Zr) clusters.^16,24,28 However, the other caged geometries 15b, 15c, and 15d, with the Ho atom being completely enclosed by silicon frames, have some dangling bonds being terminated by the enclosed Ho atom. The 15b and 15b geometries are obtained from the optimized C_5v geometry and 15d is yielded after a silicon atom is surface-capped onto the 14c isomer. However, the stability of the irregular basket-like 15b and 15c structures along with the ring-like 15d structure is quite weakened as compared to that of 15a. Additionally, the Ho encapsulated within the heptagonal prism is an important block which is appropriate for the building block of quasi-one-dimensional Ho-doped silicon semiconductive nanomaterial. Furthermore, the critical size of the Ho atom-doped silicon clusters is larger than that of the TM atom-doped silicon clusters.^29,30

Nakajima et al. investigated the adsorption reactivity of anionic HoSi_n⁻ and the experimental results suggest that the Ho is still incompletely encapsulated into the Si frame when the size of the Si cage swells to 16 atoms.³ One year later, photoelectron spectroscopy was utilized to study a variety of LnSi_n⁻ cluster anions (Ln = Yb, Eu, Sm, Gd, Ho, Pr; 3 ≤ n ≤ 13); for larger clusters resembling the Ho category it is predicted that the critical size of silicon clusters encapsulating Ho is localized at n > 11, approximately at about n = 11–15;⁵ interestingly, this suggestion is in good agreement with our calculated results of critical size n = 15. Moreover, the XLYP method in combination with effective ECP can give reasonable results of bond length and frequency. Consequently, our calculated results provide support for our selected methods.

HoSi₁₆. For the HoSi₁₆ clusters, the most perfect fullerene-like stable 16a isomer and the amorphous low-lying 16b and 16c isomers are optimized. The fullerene-like HoSi₁₆ 16a geometry with unbalanced Si bonds is different from that of the TM@Ge₁₆ isomer¹⁷ and the bimetal Pd₂-doped silicon frame.²⁵ Additionally, another 16b isomer, which is obtained after the Si atom is capped onto the pentagonal prism 15a isomer, is higher in total energy than the fullerene-like 16a structure with a calculated frequency of 36.3 cm⁻¹. The 16b isomer can be described as the Ho atom being encapsulated by two connected Si₁₀ units. Furthermore, the 16b isomer is a new building block for forming new silicon clusters encapsulating Ho. After a Si atom is added to the surface of the 15d isomer, the sandwich-like octagonal prism 16c with Ho being localized within the silicon frame is obtained. Different from the 15d isomer in which Ho is encapsulated, the Ho atom in 16c isomer is encapsulated within the centre of an octagonal prism and interacts with 16 silicon atoms with inequivalent bond lengths, and tries to saturate the dangling bonds of the silicon atoms. As far as the 16d isomer is concerned, its geometry can be described as two silicon atoms being separately surface-doped on the deformed heptagonal prism HoSi₁₄ frame; the 16d isomer is higher in total energy than the 16a isomer by 1.05 eV, reflecting the fact that the deformed octagonal prism 16c is more stable than the 16d isomer. As a consequence, the layered sandwich-like 16c and 16d isomers are not the dominant geometries because they are less stable than the cage-like 16a geometry.

HoSi₁₇. With respect to the HoSi₁₇ geometries, all optimized HoSi₁₇ structures are shown as the sealed Si₁₇ cage encapsulating Ho. The most stable fullerene-like HoSi₁₇ 17a isomer is generated, which can be described as one silicon atom being capped onto the 16a isomer. A low-lying 17b isomer can be found as a stable structure and its total energy is higher than that of the 17a isomer by 0.096 eV. As shown in Fig. 1, when one silicon atom is capped between the 1^st and 7^th silicon atoms of the HoSi₁₆ 16a isomer, a sealed caged 17b HoSi₁₇ cluster is obtained. Another stable 17c isomer with the total energy being larger than the 17b isomer can be described as the Si-capped tetragonal prism and it is different from the 17b isomer in that the Si–Si dimer is symmetrically distributed at each side of the tetragonal prism. Additionally, one silicon capped octagonal prism 17d structure is found as the stable structure; however, its total energy is higher than that of the other isomers. This geometry is yielded from 16c. The 17d isomer is less stable than the 17a isomer because the Ho-coordinated silicon atoms in the former are bigger than those of the latter. Hence, the stability of the caged Si_n clusters encapsulating Ho is related to the number of Ho-coordinated silicon atoms.

HoSi₁₈. In this size, five stable cage-like HoSi₁₈ isomers, 18a, 18b, 18c, 18d, and 18e, are obtained. The 18a isomer, the lowest-energy isomer and the most stable structure, is formed with one Si atom being surface-capped onto the deformed HoSi₁₇ 17b isomer whereas the 18b isomer is described as one Si atom being surface-capped onto the 17c isomer. The 18b isomer is higher in total energy by 0.553 eV than the 18a isomer. The 18c isomer is generated by capping one silicon atom on the surface of 17c while the 18d isomer can be seen as a rhombus being surface-capped on the deformed sandwich-like heptagonal prism D_7h 14c. Furthermore, the sandwich-like geometry is higher in energy than the two connected Si₁₀ units’ (16b-like) geometry. The 18e geometry deforms 18a geometry with the encapsulated Ho atom being localized at a different position; in other words, the encapsulated Ho atom in the 18a and 18e isomers interacts with different silicon atoms and has different coordination numbers; thus, the encapsulated Ho atom tries to terminate dangling bonds of different silicon atoms.

HoSi₁₉. As far as the HoSi₁₉ 19a isomer is concerned, the geometry can be described as the Ho atom being encapsulated into the slightly distorted cage-like Si₁₉ frame. The formed 19a isomer is composed of the two Si₁₀ units and it is generated after a silicon atom is surface-added onto the 18a isomer. From the calculated total energies listed in Table 1, it is obvious that the 19a geometry is the lowest-energy isomer. The distorted C_2v HoSi₁₉ 19b isomer is obtained after the Si atoms are surface-capped onto the distorted sandwich-like D_6h HoSi₁₂ frame. However, the 19a isomer is more stable than the 19b isomer. Guided by the above investigations of the 18d isomer, the striking 19c and 19d isomers are also considered. The C_2v 19c isomer is depicted as the Ho atom being inserted into the slightly deformed heptagonal prism Si₁₄ frame, together with a quadrangular pyramidal Si₅ unit being capped on the side of the Si₁₄ frame. It should be mentioned that the 19c isomer is higher in total energy than the 19a and 19b isomers by 2.65 and 0.11 eV, respectively. As a consequence, the layered 19c structure is not the most stable building block for the formation of the new material. The 19d isomer is depicted as the Ho atom being inserted into the layered sandwich-like octagonal prism HoSi₁₆ unit. Interestingly, this structure is lower in stability than the 19a isomer and is not selected as the ground state and building block of new materials. Therefore, the sandwich-like 19c and 19d isomers are not the dominant geometries which are examined by the calculated total energies.

HoSi₂₀. Starting from various initial structures, the HoSi₂₀ cluster is calculated. Three isomers are identified for the Ho-doped Si₂₀ clusters, which are the distorted closed cage-like D_2d 20a isomer with the enclosed Ho atom acting as the centre of a symmetrical axis, the layered heptagonal prism Si₁₄ unit with the enclosed Ho atom being localized at the centre site and a book-like Si₆ unit being surface-capped on the heptagonal prism, 20b, and the seriously distorted closed cage-like hexagonal prism D_6h 20c isomer, respectively (Fig. 1).

Two remarkable fullerene-like silicon isomers encapsulating Ho, namely 20a and 20c, are formed. The 20a isomer is obtained from the 19a isomer. It has a perfect fullerene-like geometry with Ho being localized at the centre of the Si₂₀ frame. The Ho atom in 20a tries to interact with more silicon atoms and terminate the dangling bonds of the silicon atoms. Hence, the perfect closed fullerene-like 20a structure has enhanced stability and is appropriate for the building block of optical and optoelectronic materials. Interestingly, the encapsulated Ho atom in the 20a isomer, which has geometry similar to that of the Pd₂-doped Si₂₀ isomer,²⁵ behaves as an acceptor of charges which is due to the tendency of Ho(6s²4f¹¹) to attain a completely filled 5d¹⁰ and 4f¹⁴ configuration.^29,30 The 20c isomer is yielded after two silicon atoms are symmetrically capped on two sides of the 18b cluster. According to the calculated total energies listed in Table 1, it is obvious that 20c is higher in total energy than 20a by 3.95 eV. In addition, the 20b isomer is generated after a silicon atom is added on to the surface of the 19c isomer; moreover, the 20b isomer is less stable than the closed cage-like 20a cluster with a total energy difference of 3.34 eV.

3.2. Relative stabilities

In order to investigate the relative stabilities of the most stable HoSi_n (n = 12–20) clusters, the atomic averaged binding energies for the HoSi_n clusters (E_b(n)) and the fragmentation energies (D(n,n − 1)) with respect to removal of one Si atom from the most stable HoSi_n clusters are investigated (Fig. 2 and Table 2). The atomic averaged binding energies and fragmentation energies of the HoSi_n (n = 12–20) clusters are defined as:

E_b(n) = [E_T(Ho) + nE_T(Si) − E_T(HoSi_n)]/(n + 1)

D(n,n − 1) = E_T(HoSi_n−1) + E_T(Si) − E_T(HoSi_n)

where E_T(HoSi_n), E_T(HoSi_n−1), E_T(Si), and E_T(Ho) denote the total energies of HoSi_n, HoSi_n−1, Si, and Ho clusters, respectively. The calculated results of the most stable HoSi_n (n = 12–20) isomers are plotted as the curves of E_b and D(n,n − 1) against the corresponding number of the Si_n atoms (Fig. 2 and Table 2). It should be mentioned that the features of the size-evolution are intuitively viewed and the peaks of the curves correspond to those clusters having enhanced local stabilities.

Fig. 2 The calculated averaged atomic binding energies (a) and fragmentation energies (b) for the most stable HoSi_n (n = 12–20) clusters.

Table 2 The calculated fragmentation energies and the atomic averaged binding energies of the most stable HoSi_n (n = 12–20) clusters (unit: eV)

System	12	13	14	15	16	17	18	19	20
Eb	2.78	2.80	2.80	2.85	2.90	2.88	2.95	2.94	3.01
D(n,n − 1)		3.12	2.81	3.53	3.74	2.55	4.17	2.87	4.36

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The atomic averaged binding energies of the HoSi_n clusters are calculated and are listed in Table 2 and displayed in Fig. 2a; the curve generally shows the tendency for increased binding energy as the size of n is increased. As seen from Fig. 2a, it is noticed that three peaks with n = 16, 18, and 20 are found, reflecting that the predicted relative stabilities of the HoSi_n (n = 16, 18, and 20) clusters have stronger local stabilities as compared with the others.

According to the calculated fragmentation energies shown in Fig. 2b and Table 2, three remarkable peaks at n = 16, 18, and 20 for the HoSi_n (n = 12–20) clusters are found, showing that the corresponding clusters have stronger relative stabilities and have large abundances in mass spectroscopy as compared to the others. Moreover, the order of stabilities is HoSi₂₀ > HoSi₁₈ > HoSi₁₆; this finding is in good agreement with the calculated atomic averaged binding energy. Interestingly, the Ho-doped silicon clusters have different magic numbers of stabilities as compared with the single transition metal doped silicon clusters.³⁰ The 20a isomer in terms of the calculated fragmentation energies and averaged atomic binding energies shows the strongest geometrical stability, reflecting that the closed fullerene-like 20a geometry is the preferred structure. According to the calculated total energies of the HoSi₂₀ 20a and HoSi₁₆ 16a isomers, it is noticed that the 20a and 16a isomers can act as the building blocks for nanomaterials.

3.3. Charge-transfer mechanism

The calculated atomic polar tensor-based charges (APT) are tabulated in Table 1 and are displayed in Fig. 3. According to the calculated APT data for the Ho atom in the HoSi_n (n = 12–20) clusters, all the APT values of Ho are negative except for the APT value of the Ho atom in the 13a isomer (Table 1), showing that the electronic charges in the corresponding clusters transfer from the Si_n frames to the Ho atom; and the unsaturated d orbitals of the Ho atom obtain the electrons from the Si atoms because the f orbitals are not involved in chemical bonds. These results coincide with the previous findings for ReSi_n and WSi₁₅.^13,15 As is the general trend, the APT populations of Ho are first decreased to a certain value and then increase. The two peaks at n = 16 and 19 correspond to the clusters with large charge-transfers and the charge-transfer of the HoSi₁₆ cluster is the biggest one. It should be pointed out that the encapsulated Ho atom in the most stable HoSi_n (n = 12–20) clusters obtains more charges from its surroundings than the surface-capped Ho atom, and that the encapsulated Ho atom has a tendency to interact with more silicon atoms with inequivalent bond lengths and tries to terminate the dangling bonds of the silicon atoms. Therefore, the doped Ho atom plays very important roles in determining the stabilities of the Ho doped Si_n (n = 12–20) clusters.

Fig. 3 Size dependence of the APT of the most stable HoSi_n (n = 12–20) clusters.

3.4. HOMO and LUMO properties

The HOMO–LUMO gaps of the HoSi_n clusters are tabulated in Table 1. Based on our calculated results, the HOMO–LUMO gaps for the most stable HoSi_n (n = 12–17) clusters are about 1.5 eV while the HoSi₁₆ cluster (1.95 eV) has the biggest HOMO–LUMO gap, indicating that the most stable HoSi_n (n = 12–17) clusters would be effective semiconductor materials (Fig. 4). In other words, the HoSi₁₆ cluster is the strongest one, in terms of chemical stability, of all the HoSi_n (n = 12–17) clusters. Therefore, having a stronger relative stability and chemical stability, the HoSi₁₆ cluster can be seen as the most suitable building block and can be selected as a candidate for novel nanomaterials. As far as the HoSi₂₀ cluster is concerned, the HOMO–LUMO gap is 1.45 eV which is smaller than that of the HoSi₁₆ (1.95 eV) cluster and therefore the HoSi₁₆ cluster is stronger in terms of chemical stability than the HoSi₂₀ cluster. Furthermore, we can predict that the HOMO–LUMO gaps for the HoSi₂₀ and HoSi₁₆ clusters can be detected experimentally. Consequently, the HoSi₁₆ and HoSi₂₀ clusters are stable in ionization and dissociation processes and contribute to forming the most preferred Ho-doped silicon nanoclusters, and make them more attractive for the cluster-assembled rare earth doped semiconductive nanomaterials with novel optical and optoelectronic properties.

Fig. 4 Size dependence of the HOMO–LUMO gap of the most stable HoSi_n (n = 12–20) clusters.

The contour map of the HOMO for the most stable isomers (HoSi₁₆, HoSi₁₈, and HoSi₂₀) is displayed in Fig. 5. It can be seen from Fig. 5 that some π-type and σ-type bonds are formed among the Si_n atoms. The encapsulated Ho atom in the HoSi₁₆ isomer interacts with 16 silicon atoms simultaneously and terminates the dangling bonds of the silicon atoms. As far as the HOMO property of the HoSi₁₆ cluster is concerned, the π-type bonds are formed among the Si_n atoms, which are distributed beside the Ho atom, and correspond to the localized π-type bonds. Furthermore, the encapsulated Ho atom obtains more charges from its silicon atoms and interacts with silicon atoms by relatively d–s σ-type bonds. From the contour maps of the HOMOs for the HoSi₁₆ HoSi₁₈, and HoSi₂₀ clusters, it is obvious that the encapsulated Ho atom is surrounded by silicon atoms and only a few Si–Si bonds contribute to the π-type bonds because the σ-type bonds are the dominant Si–Si bonds in HoSi₁₆ HoSi₁₈, and HoSi₂₀ clusters.

Fig. 5 The contour maps of the HOMO for the most stable 16a, 18a, and 20a isomers.

3.5. Dipole moments

The dipole moments, which can be used to reflect the global geometry of the HoSi_n (n = 12–17) clusters, are calculated. The calculated results for the most stable Ho doped Si_n clusters are tabulated in Table 1 and shown in Fig. 6. As seen from Fig. 6, it is apparent that the curve shows an oscillating tendency as the size of the silicon cluster increases. The three local minima are for HoSi₁₆, HoSi₁₈, and HoSi₂₀ clusters, corresponding to dipole moments of 0.03, 0.31, and 0.0 Debye, respectively. HoSi₁₆ and HoSi₂₀ clusters are completely global geometries.

Fig. 6 The calculated dipole moments of the most stable HoSi_n (n = 12–20) clusters.

4. Conclusions

The HoSi_n (n = 12–20) clusters with different spin states have been systematically investigated at the level of the DFT with the hybrid exchange and correlation (X3LYP) functional in combination with the 6-31G basis sets for the Si atoms and the Large-core Stuttgart effective core potentials (ECP56MHF) for the Ho element. The total energies, growth-pattern, equilibrium geometries as well as the APT charges of the HoSi_n (n = 12–20) clusters are calculated.

(1) The relative stabilities in terms of the calculated atomic averaged binding energies and fragmentation energies are discussed in detail. The calculated results reveal that the cake-like HoSi_n (n = 16, 18, 20) clusters have enhanced stabilities as compared with the others and HoSi₂₀ is the most stable one. The order of stability is predicted to be HoSi₁₆ < HoSi₁₈ < HoSi_20. Interestingly, the calculated dipole moments of the cake-like HoSi_n (n = 16, 18, 20) clusters are a lot smaller than that of the others, which correspond to the global minima; furthermore, the dipole moments for HoSi₁₆ and HoSi₂₀ are 0.03 and 0.00 Debye, respectively.

(2) The highest occupied molecular orbital – lowest unoccupied molecular orbital (HOMO–LUMO) gaps of the HoSi_n (n = 12–17) clusters are calculated. The calculated values for the HoSi_n (n = 12–17) clusters are above 1.55 eV while HoSi₁₆ has the largest value (1.95 eV), reflecting that the HoSi_n (n = 12–17) cluster is a good semiconductor material, especially the HoSi₁₆ cluster.

(3) According to the calculated APT charges of the Ho atom in the HoSi_n (n = 12–20) clusters, the contribution of charge-transfer to the stability of the HoSi_n clusters is briefly discussed, manifesting that the charges in the HoSi_n clusters transfer from the Si atoms to the Ho atom, and the charge-transfer in the HoSi₁₆ cluster is the biggest one and has enhanced stability.

(4) The Ho atom in the lowest-energy HoSi_n (n = 12–14) isomers occupies the surface site while the optimized geometries show that the rare earth Ho atom in the HoSi_n (n ≥ 15) clusters is completely encapsulated into the Si frame. Furthermore, the critical size of the clusters required for the enclosed Ho atom to completely fall into the Si_n (n = 12–20) frames is predicted to be n = 15; this finding is in good agreement with experimental results.⁵

Acknowledgements

This work is supported by the Natural Science fund of China (11179035, 10979048, 10875126), the 973 fund of the Chinese Ministry of Science and Technology (2010CB934504), the Innovation Program of Shanghai Municipal Education Commission (14YZ164 and 12YZ185) as well as Physical electronics disciplines (no. 12XKJC01).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c4ra11828f

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