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Strong Electron Correlations Determine Energetic Stability and Electronic Properties of Er-Doped Goldberg-Type Silicon Quantum Dots Pavel V. Avramov,*,†,‡ Dmitri G. Fedorov,§ Stephan Irle,|,⊥ Alexander A. Kuzubov,†,‡ and Keiji Morokuma⊥ L.V. Kirensky Institute of Physics SB RAS, Krasnoyarsk 660036, Russia, Siberian Federal UniVersity, 79 SVobodniy aV., Krasnoyarsk 660041, Russia, RICS, National Institute of AdVanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan, Institute for AdVanced Research and Department of Chemistry, Nagoya UniVersity, Nagoya 464-8602, Japan, and Fukui Institute for Fundamental Chemistry, Kyoto UniVersity, 34-3 Takano Nishihiraki, Sakyo, Kyoto 606-8103, Japan ReceiVed: May 28, 2009; ReVised Manuscript ReceiVed: July 22, 2009
Atomic and electronic structures of Goldberg-type silicon quantum dots and their endohedral erbium complexes were studied using ab initio and plane wave pseudopotential density functional and Møller-Plesset manybody perturbation theories. During atomic structure optimizations, the erbium ions occupy mass centers inside the central hollows of quantum dots of different symmetries. It was found that strong electron correlations within the Er 4f shell taken into account by empirical pseudopotential and post-Hartree-Fock perturbation approaches are responsible for strong binding of Er ions to quantum dots. We elucidate the effects of symmetry and discuss theoretical results in comparison to available experimental data. I. Introduction Er-doped silicon in different forms has been attracting much interest because of its potential application as an efficient semiconductor light emitter.1-13 From the technological point of view, Er-doped nanocrystalline silicon (nc-Si) embedded into silica (nc-Si:Er/SiO2, with effective nc-Si core sizes of 1.5-5 nm) in particular is a very promising material to produce semiconductor light emitter devices.6-8,13 The photoluminescence (PL) of nc-Si:Er/SiO2, driven by the 4I15/2 r 4I13/2 f-f transition in the [Xe]f11 configuration of Er3+, is thought to originate from the recombination of photogenerated carriers in nc-Si and the subsequent energy transfer to Er3+. Complexes of d-transition metals (Hf, Ta, W, Re, Ir, etc.) encapsulated in small Sin clusters have also been synthesized using an ion trap and theoretically investigated using ab initio and density functional theory (DFT) methods with the local density approximation (LDA).14,15 Analysis of experimental spectroscopic data6-8,13,16-18 implies that each Er ion is included in one nc-Si quantum dot of 1.5-5 nm size and does not form covalent Er-O chemical bonds with the silica environment. The most probable interpretation of these facts is that erbium penetrates into the hollows of nc-Si, which is covered outside by silica; yet the precise atomic structure of such Er-nc-Si/SiO2 conglomerates is experimentally entirely unknown. In this work, we attempt to give answers based on first principles DFT and ab initio second-order Møller-Plesset perturbational theory (MP2) calculations to the following questions: (a) What is the molecular and electronic structure of Er-doped nc-Si quantum dots? (b) * Corresponding author. E-mail:
[email protected]. Telephone: (7)-391249-4456. † L.V. Kirensky Institute of Physics SB RAS. ‡ Siberian Federal University. § RICS, National Institute of Advanced Industrial Science and Technology. | Institute for Advanced Research and Department of Chemistry, Nagoya University. ⊥ Fukui Institute For Fundamental Chemistry.
What is the nature of the electronic correlations within the Er 4f shell? (c) What is the origin of stability for these species? II. Methods of Electronic Structure Calculations In our theoretical methodology, we followed a multifold approach: The VASP19,20 quantum chemical code was employed in LDA calculations on the basis of the Vanderbilt Ultra-Soft pseudopotential projector augmented plane wave method (PP PAW),21,22 while the Gaussian23 program was employed in B3LYP24,25 and MP226 calculations utilizing atom-centered Gaussian-type basis sets. The reasons for using these different quantum chemical methods are (i) to elucidate the nature and role of the strong electron correlations (SEC), which are the driving force in many systems involving transition metals,27-29 and can be taken into account by PP PAW (see below) and MP2 but are inadequately described by B3LYP and (ii) to study the partial density of Er 4f electronic states, which can be defined in B3LYP but not in PP PAW and MP2. To construct total and partial density of states (TDOS in PP PAW and PDOS in B3LYP, respectively), we applied Gaussian broadening with a smearing width of 0.1 eV for the corresponding electronic occupation numbers of molecular orbital levels. For B3LYP and MP2 calculations, the 3-21G* basis set was employed for all Si and H atoms. For Er, we used the Stuttgart relativistic small core (RSC) 1997 basis set30 with an effective core potential for [Ar]3d10 (SRSC97-ECP) and a large, flexible basis set for 4s24p65s25p64d106s25d04f12 valence electrons. During SCF iterations in Gaussian, we applied a 1 × 10-6 hartree self-consistency threshold for the total energy, and geometry optimizations were performed until the gradient was smaller than 10-4 hartee/bohr. All geometries were fully optimized in C1 symmetry to avoid converging to higher energy solutions for the wave function imposed by symmetry. The B3LYP and MP2 relative energies were calculated by taking into account the basis set superposition error (Boys-Bernardi counterpoise scheme). For silicon containing systems, the MP2 approach gives isomer relative energies and heights of potential
10.1021/jp904996e CCC: $40.75 2009 American Chemical Society Published on Web 08/18/2009
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barriers at transition states close to the results of the much more sophisticated coupled cluster method.31 For PP PAW calculations, the Vanderbilt Ultra-Soft pseudopotential with an Er effective core potential for [Xe]4f11 configuration was used. The Vanderbilt Ultra-Soft pseudopotentials significantly reduce the maximal kinetic energy cutoff without significant loss of accuracy. We thus used an Ecutoff energy equal to 250 eV. Geometry optimizations were carried out until the forces acting on all atoms became lower than 0.001 eV/Å while applying a 1 × 10-4 eV threshold as self-consistency energy criterion during individual electronic structure calculations. The strong electron correlations in the Er/Si systems32,33 were effectively simulated by the excitation of one Er 4f electron from the Er 4f12 shell to the valence Er 5d shell. This approach allows us to model the photoemission active Er 4f11 configuration of Er3+ and to compute binding energies in a consistent manner, which would not be possible if LDA+U34 was used, another commonly employed empirical approach allowing the simulations of strong electron correlation effects in DFT calculations. To calculate the binding energy of Er to the silicon clusters, we found the total energy of all constituting parts should be consistently calculated in the same manner. Strong electron correlations within the Er 4f shell in erbium silicides are caused by Coulomb interactions of the localized erbium f electrons and silicon p subband, and in LDA+U calculations a finite value of U is necessary to account for this interaction.32,33 A free erbium atom cannot be calculated within the same LDA+U approach because U is equal to zero for atoms because of the very definition of the LDA+U approximation. The free Er atom and Er-doped quantum dots were computed by B3LYP and MP2 in their triplet electronic ground state. Note that the Er ground state was experimentally determined to be 3H; it was found that for Er/Si systems, the triplet state has the lowest energy (electronic ground state of Er3+ is 4I, which is coupled with the silicon cluster ground state possessing three excess electrons to a triplet state in Er@GQD). It was found that in B3LYP calculations the Er atom has a Mulliken charge of about +2 with a small mixture of Er 6s and Si p atomic orbitals corresponding to Er2+. However, in PP PAW, the Er atom is formally in the Er3+ state, where the entire Er 5d partial density of states is localized in the vacant state energy region, whereas the atomic Er 6s state exhibits only a small contribution to the intensity of the embedded electron state at the Fermi level. In the following paragraph, we wish to comment on the relationship between the impurity Anderson model35 and the MP2 method for the description of electron correlation. The impurity Anderson model has been successfully used for decades to study the strong electron correlations in the systems with localized electronic states (for example see extended review36). In terms of secondary quantization, the Hamiltonian of the model can be written in the following form H
A
(
1 K 2
)
∑ε c
+ p pσcpσ
p,σ
∑ ∑f
+
∑ε
f + kσ fkσ fkσ
k,σ
+ + kσ′ fkσ fkσ fkσ′
k
+
∑V
p,k,σ
∑ ∑f k*l
σ*σ′
1 J 2
∑ ∑f k,l
+ + fkσ + fkσ cpσ)+ (cpσ
f pkσ
)
+ + lσ′ fkσ fkσ flσ′
σ,σ′
k,l
σ,σ′
+
-
+ + lσ fkσ fkσ flσ
σ
k*l
(1) + cpσ
+ fkσ
where cpσ and fkσ and and are annihilation and creation operators of the pth valence (c) and kth strongly correlated f
electrons (f), respectively. Following typical notations for f Anderson-type models, K, J and Vpkσ are repulsive, exchange and hybridization parameters within and between valence and strongly correlated f state subspaces.35 The sum of the first and + f + cpσ + ∑k,σεkσ fkσfkσ corresponds to the the second terms ∑p,σεpcpσ f are one-electron Hartree-Fock (HF) energy, where εp and εkσ eigenvalues of the valence and f states, respectively. The parameters of the model can be successfully calculated using an ab initio technique.37 The third term of the Hamiltonian involves two states and describes a coupling of the valence and highly correlated f states through electron hopping (excitation) between occupied and virtual orbitals. Actually, this term is responsible for the competition of the f n and f n-1 configurations. The remaining two terms couple two occupied f states with the same or different spins through Coulomb and exchange interactions and also correspond to HF. Howeverd, the MP2 correlation energy describes the coupling of two occupied and two virtual orbitals.26 The effective MP2 Hamiltonian, describing the correlation energy, in the second quantization form38,39 has the following form
H
MP2
)
(∑
∑
)
1 (wr|xs)Ewrxs × 2 rwsx 1 trwErw + trwsxErwsx 2 wr wrxs
(w|h0 |r)Ewr +
rw
(∑
∑
)
(2)
† awσ where the unitary group operators are given by Erw ) ∑σarσ † † and Erwsx ) ∑σ∑σ′arσasσ′axσ′awσ; w and x are occupied; r and s are vacant one-electron states (orbitals); and trwsx and trw are amplitudes in the MP2 theory. The MP2 energy38 is calculated by adding the MP2 correlation energy to the RHF energy, and can be written as
EMP2 ) 〈R|H|R〉 + 2
∑ trw〈w|h0|r〉 + ∑ w,r
〈wr|xs〉 ×
r,s,w,x
(2trwsx - trxsw) (3) where h0 is the zero-order one-electron Hamiltonian, H is the all-electron Hamiltonian, and R is the HF wave function. Equation 3 bears some similarity to eq 1. By comparing eqs 1, 2 and 3, we notice that in the Anderson Hamiltonian of eq 1 there is a single excitation term from the strongly correlated space into the valence space, which corresponds to a part (because it applies to limited subspaces, whereas in MP2 to full spaces) of the corresponding single-particle term in MP2; this term in MP2 is zero38 for canonical orbitals because the zero-order Hamiltonian is diagonal, while in the Anderson model the potential V is not. If one used some analogue of the noncanonical basis used in the Anderson model, there would be a similar term in the MP2, with the difference that this V in MP2 would strictly be ab initio, whereas in the Anderson model it can be constructed by any external means (i.e., fitted to the experiment). Thus, we summarize the comparison of the Anderson and MP2 models as follows. Both share the very important physical notion of treating the effect of populating the valence space by the excitation of the strongly correlated electrons (i.e., transfer of f electrons to the virtual orbitals of ligands), and the difference is that the Anderson model only considers single-electron excitations, whereas MP2 only includes double excitations. III. Results and Discussion Now, we present molecular and electronic structure calculations of erbium-doped nc-Si quantum dots. The addition of
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Figure 1. Structures of Er-doped silicon quantum dots (optimized with PP PAW). Hydrogen atoms are hidden. The inner silicon core is represented by green sticks, outer silicon atoms are purple, and Er is red. Symmetry (point group) is shown for the inner core Si20-Si28, followed after a slash with the point group for the whole system.
TABLE 1: Structure and Stability of Er/Si Clustersa Si Ecoh (kcal/mol)
∆EBG (eV)
Er Ecoh (kcal/mol)
system
core
size (nm)
PPPAW
B3LYP
PPPAW
B3LYP
MP2
Si20Si80H60 Si24Si96H72 Si26Si104H78 Si28Si112H84 Er@Si20Si80H60 Er@Si24Si96H72 Er@Si26Si104H78 Er@Si28Si112H84
Ih D6d D3h Td Ci D2d D3h Cs
1.3 1.4 1.6 1.5 1.3 1.4 1.6 1.5
2.041 1.613 1.672 1.760 0.924 1.043 1.224 1.445
3.428 2.958 3.012 3.125 1.199 1.506 1.843 2.052
-87.1 -86.4 -86.3 -86.3
-101.3 -100.7 -100.6 -100.7
-121.1 -121.6
PPPAW
-76.4 -67.6 -60.4 -49.1
B3LYP
22.1 -0.7 0.9 -3.3
REr-Si (Å) MP2
-131.6 -145.1
PP PAW 3.285-3.323 3.461-3.930 3.543-4.435 3.980-4.148
a B3LYP/3-21G* and MP2/3-21G* data are presented for triplet states. ∆EBG is the energy of the band gap, REr-Si is the Er-Si distance, and Si Er Er Ecoh and Ecoh are cohesion energies of silicon and erbium atoms, respectively. The Ecoh values were calculated taking into account basis set superposition error (BSSE).
several 〈111〉 silicon layers to the bare central Sim (m ) 20 + k, k ) 0, 4, 6, 8...) fullerene-like hollow cages produces Si Goldberg quantum dots (GQD).40,41 Because of enough empty space inside the central Si20+k cages, the GQDs allow one to create endohedral complexes with guest atoms or molecules (Figure 1).40,42 Perfect symmetries of parent GQDs are determined by the inner Si20+k cores. For example, the icosahedral dot Si20Si80H60 with one 〈111〉 external layer has a dodecahedral Si20 core.41 Larger dots can be made by adding external silicon layers to the inner silicon core.40 Electronic structure calculations have shown that the first members [Si20Si80H60 (Ih), Si24Si96H72 (D6d), Si26Si104H78 (D3h), and Si28Si112H84 (Td)] of the GQD family are energetically preferable among all possible silicon quantum dots with effective size up to 3-5 nm.40,41 To model erbium-doped GQDs, we calculated the molecular and electronic structure of the endohedral complexes Er@GQD of erbium ions encapsulated within nc-Si Goldberg quantum dots Si20Si80H60 (Ih), Si24 Si96H72 (D6d), Si26Si26Si104H78 (D3h), and Si28Si112H84 (Td) (the symmetries of the inner cores are shown in parentheses, Figure 1), with surface dangling bonds being saturated by hydrogen atoms. All parent SinHm Goldberg-type silicon quantum dots display high energetic stability at all PP PAW, B3LYP/3-21G*, and MP2/3-21G* levels of theory (Table 1) with similar GQD H Si cohesion energies ESi coh ) (Etot - mEbond - nE )/n of the silicon GQD atoms, where Etot is the total energy of a SinHm quantum dot, ESi is the total energy of the free silicon atom in its 3P electronic H is ground state at the corresponding level of theory, and Ebond H the energy of a hydrogen atom in the SiH4 molecule (Ebond ) SiH4 - ESi)/4. Actually, the silicon cohesion energy is the (Etot binding energy per silicon atom, and for all GQDs, it is
approximately equal to -86, -101, and -122 kcal/mol at the PP PAW, B3LYP/3-21G*, and MP2/3-21G* levels, respectively. Electron correlations within the Er 4f shell of all endohedral complexes lead to a distortion of the silicon cages with significant symmetry lowering (Table 1). The Er ion always occupies the center of mass of the clusters (Figure 1). For all Er-doped quantum dots, the B3LYP calculations predict either Er (-3.3 kcal/ repulsive (22.1 kcal/mol) or small attractive Ecoh mol) cohesion energies for the erbium ions, which is determined Er Er@GQD GQD Er Er@GQD ) Etot - Etot - Etot , where Etot is the total as Ecoh Er is the total energy of the Er@GQD complex (Table 1) and Etot Er values were calculated taking energy of the Er atom. All Ecoh into account the basis set superposition error (BSSE). In most cases the BBSE correstions were found to be smaller 3 kcal/ mol. The difference between erbium cohesion energies ∼100 kcal/ mol (4.3 eV) at B3LYP (without SEC) and PP PAW (with SEC) levels of theory (Table 1) is on the same order as the Hubbard U constant (6.2 eV) typically applied in the LDA+U method.32,33 In other words, SEC determines the large binding energies of the erbium ions in the central hollows of Goldberg-type quantum dots. Direct inclusion of dynamical electronic correlations using MP2 theory for the Hartree-Fock (HF) ground state Er 4f12 Er reference configuration leads to very negative Ecoh values (-131.6 and -145.1 kcal/mol for Er@Si100H60 and Er@Si120H72 systems, respectively, Table 1), while the approximate treatment of SEC effects in the PP PAW method leads to significantly Er smaller Ecoh values (the erbium cohesion energies vary from -76.4 kcal/mol for Er@Si100H60 to -49.1 kcal/mol for Er@Si140H78).
Er-Doped Goldberg-Type Silicon Quantum Dots
Figure 2. (a) TDOS of Goldberg quantum dots (GQD), obtained using PP PAW (red dashed lines) and B3LYP/3-21G*/SRSC-97-ECP levels of theory (black solid lines). (b) TDOS (PP PAW) of the Er@GQD endohedral complexes (black solid lines) and PDOS of Er 4f (B3LYP, red dashed lines), whose intensity was magnified by a factor of 20. Spectroscopic features due to the Er 6s and Er 5d states are shown by arrows.
The experimental data (for example, see ref 1) as well as preliminary CASSCF calculations of small Er/SinHm clusters43 clearly demonstrate the single configuration nature of the Er/Si system wave functions corresponding to the Er 4f11 configuration. The insertion of Er inside the GQDs slightly decreases (approximately by 0.1) the weight of the reference HF wave function in MP2 for the Er@GQD systems, indicating the mainly single-configuration nature of wave functions of Er/SinHm systems. In other words, in MP2 the reference configuration is Er 4f12 (partial Mulliken population of the Er 4f states is approximately 11.9 electrons) with small electron transfer to the ligand subspace. The Er 4f states are strongly spatially and energetically localized orbitals, staying apart from the Si p subbands. Variation of the number of active orbitals in the MP2 space of the Er@GQD complexes leads to consecutive exclusion of Er 4f or Si p electrons. It was shown that an exclusion of one Er 4f electron from active MP2 space reduces the binding energy of the erbium ion to silicon dots by 28 kcal/mol on the average, whereas an average energetic effect of exclusion of an electron from an Si p subband is 13 kcal/mol (15 kcal/mol smaller). A rough estimate shows that MP2 dynamical correlations within Er 4f electrons determine the energetic stability of the Er/Si systems (15 kcal/mol × 12 ) 180 kcal/mol). As a result, MP2 overestimates the Er binding energy by ∼15 kcal/mol due to the presence of an extra electron at the Er 4f shell and exclusion of the electron from the Si p valence subband. The total (TDOS) and partial (PDOS) Er 4f density of states of the parent GQDs and Er@GQD complexes obtained at the PP PAW and B3LYP levels are presented in Figure 2. The corresponding band gaps are given in Table 1. The TDOS of the valence bands of the parent GQDs obtained at different levels of theory have similar shapes. The PP PAW method, nevertheless, gives sharper peaks, keeping the same or close energy positions with band gaps ∼1.4 eV smaller than those of B3LYP. The GQD symmetry determines the shape and energy splitting of the TDOS peaks. The Ih Si20Si80H60 dot has a sharp main
J. Phys. Chem. C, Vol. 113, No. 36, 2009 15967 peak at -7.157 eV and low-intensity maximum near Fermi level. Increasing the size of the central hollow (and consequently, the Er-Si distances) and decreasing the symmetry up to D3h results in a smearing of the main peak and its merging with the first low-intensity peak forming one wide DOS structure below the Fermi level (Figure 2b). Insertion of Er into the central hollows creates embedded levels inside the forbidden gaps. In both methods, the states are formed by mixing of the Si 3p and Er 6s orbitals (Figure 2b). The PP PAW method gives lower energetic positions of the Er 6s-derived states compared to B3LYP. Next, we discuss the differences between the four different types of GQDs. The Er atom always occupies the center, and larger hollow cores (Si24-Si28) have longer Er-Si distances, resulting in decreased stability (Table 1), because the electrostatic attraction between the negative charges on the quantum dot and the positive charges on the Er cation is reduced at longer distances. The symmetry of the central hollow produces the ligand-like field-splitting effect upon the electronic levels of Er, which is observed in Figure 2b; PDOS of Er 4f states have more peaks with larger energy splitting for the hollows with a lower symmetry. Thus, GQDs with lower symmetry are less stable but should have a different wavelength of PL transitions, providing means for designing emitters with the desired properties. IV. Conclusions In summary, the available experimental facts concerning ncSi:Er/SiO2 species having a single Er atom per quantum dot show the importance of the absence of Er-O fingerprints in XRD structural data13,17,18 as well as the existence of caged and semicaged small silicon clusters with transition metals inside.14 Comparing them with our theoretical results, we conclude and answer our questions asked above as follows: (a) Experimentally observed Er-doped Si light emitters correspond to the Goldberg quantum dots with an Er atom occupying their central hollow (the Ih variety similar to Er@Si20Si80 is the mostly likely candidate; also, a larger GQD may be formed in the experiment). (b) The Er 4f shell is clearly affected by strong electron correlations. (c) Stability is determined by strong electron correlations caused by dynamical correlations within the Er 4f shell coupled with Si 3p subbands, which can be taken into account using the MP2 theory and to some degree in PP PAW, where one Er 4f electron is excited from the Er 4f12 shell to the valence Er 5d shell. The theoretical analysis presented in this work elucidating the properties of the experimentally manufactured Er-doped light emitters creates the basis for potential design of new materials with desired optical and magnetic properties. Acknowledgment. This work was supported by a CREST (Core Research for Evolutional Science and Technology) grant in the Area of High Performance Computing for Multiscale and Multiphysics Phenomena from the Japan Science and Technology Agency (JST) and a collaborative RFBR-JSPS Grant 0902-92107-ЯΦ. One of the authors (S.I.) also acknowledges support by the Program for Improvement of Research Environment for Young Researchers from Special Coordination Funds for Promoting Science and Technology (SCF) commissioned by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. References and Notes (1) Coffa, S.; Priolo, F.; Franzo, G.; Bellani, V.; Carnera, A.; Spinella, C. Phys. ReV. B 1993, 48, 11782.
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