Local bond-electron-energy relaxation of Mo atomic clusters and solid skins

Abstract

A combination of tight-binding theory, bond order–length–strength correlation and non-bonding-electron polarization notions, photoelectron spectrometrics, and density functional theory calculations has enabled us to examine the effect of atomic undercoordination on local bond-electron-energy relaxation pertaining to Mo(100, 110) skins and atomic clusters. This exercise has led to the following quantitative information: (i) the atomic Mo 3d_5/2 energy level located at 224.862 ± 0.004 eV shifts 2.707 eV deeper upon bulk formation; (ii) the skin local bond is subject to a 9.80% contraction; and (iii) 5.952e charge transfers from the inner to the outermost skin layer. Furthermore, the E_4s level shifts from 61.229 eV for Mo₅₉ to 61.620 eV for the Mo₁₅ cluster and the valence band undergoes a 1.057 eV upward shift. The globally positive core-level shift arises from local quantum entrapment due to bond contraction and strength gain. The densely entrapped core electrons polarize the valence electrons and hence raise the valence band energy.

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1. Introduction

Solid skins and atomic clusters have attracted much attention both experimentally and theoretically due to their fascinating chemical and physical properties which are different than those of their bulk counterparts.¹ These abnormal properties include dilute magnetism,^2,3 catalytic ability,⁴ and the creation of Dirac-fermions for topological insulators.⁵ All these properties are closely related to atomic undercoordination induced bond relaxation and the electronic structure configuration.¹ Therefore, the exploration of local electronic properties of systems induced by coordination imperfection is crucial in order to improve the understanding of their electronic structure-related properties at the atomic level, such as cohesive energies,⁶ segregation energies,⁷ heats of mixing,⁸ and charge transfer.⁹

Molybdenum (Mo) nanoclusters form a class of interesting materials used as media in ultrahigh density magnetic recording¹⁰ and as catalyst in various chemical processes, such as CVD growth of carbon nanotubes.¹¹ Considerable efforts have been made on the X-ray photoelectron spectroscopy (XPS) and density functional theory (DFT) investigation of undercoordinated Mo systems.^12–16 For example, Lundgren et al.¹² found that the 3d core-level binding energy (BE) of Mo(110) surface atom is 333 ± 10 meV lower than that of bulk atoms using XPS. Yakovkin^13,17 calculated the relaxation of Mo(111, 112) skins and observed that the surface interlayer contracts by 16.5% and 13.4% in sequence, which is associated with the density redistribution and transformation of the electronic surface states. However, the measured core-level binding energy spectrum represents the mixture of the bulk component and the skin sublayers,^18,19 and the laws governing the energetic behavior of electrons and the electronic property changes of the material in solid skins and atomic clusters remain unclear. It has been a great challenge to obtain quantitative information regarding coordination-resolved surface relaxation, binding energies, and the energetic behavior of electrons localized in solid skins and atomic clusters. Therefore, it is necessary to clarify the contribution of surface local relaxation and reveal the mechanism of the effects of coordination imperfection on the electronic structures, which should be useful for understanding their electronic structure-related properties at the atomic level.

In this work, we present our examination of the atomic undercoordination effects on local bond relaxation, binding energy and associated core-level energy shifts and valence band variation of Mo solid skins and atomic clusters based on bond order–length–strength (BOLS) correlation and non-bonding electron polarization (NEP) premises.²⁰ The consistency between quantum calculations and photoelectron spectroscopy measurements of Mo skins and nanoclusters confirms our predictions that atomic undercoordination shortens and strengthens bonds, localizes and entraps core electrons, and polarizes the valence charges, which modify the Hamiltonian, the atomic cohesive energy, and the local binding energy density, and finally result in the unusual properties of Mo skins and nanoclusters.

2. Principles and calculation methods

2.1 BOLS-TB notation

According to the BOLS correlation,¹ shorter and stronger bonds between undercoordinated atoms result in local densification and quantum entrapment of core electrons, which make the valence charges polarise. The bond length and bond strength between undercoordinated atoms are closely associated with the effective atomic coordination number (CN), which is expressed as:where C_i is the coefficient of bond contraction with z_i being the CN of an atom in the ith atomic layer, i is counted from the outermost atomic layer inward up to the third layer, d_i and E_i are the bond length and bond energy in the ith atomic layer, respectively, and E_C(z) is the atomic cohesive energy. The bond nature indicator m represents how the bond energy changes with the bond length. For most metals, m = 1. For nanoclusters, z₁ = 4(1 − 0.75/K), z₂ = z₁ + 2 and z₃ = 12, where K represents the dimensional size equal to R/d_b. R is the real size of the specimen and d_b is the bulk bond length of the corresponding material.

According to the combination of BOLS correlation and tight-binding (TB) theory,²¹ the single-body Hamiltonian is perturbed by the shorter and stronger bonds, denoted with Δ_H:

where,

Note that the shape factors τ = 1, 2, and 3 correspond to a thin plate, a cylindrical rod, and a sphere dot, respectively.

Regarding the core level energy of a material, we have mainly focused on two characteristic energies: the νth energy level of an isolated atom E_ν(0) and the bulk shift (BS) ΔE_ν(12) = E_ν(12) − E_ν(0), where z = 12 means the bulk counterpart. The former is the integral of the eigen wave function and the intra-atomic potential, and the latter is its energy shift upon bulk solid formation. Both E_ν(0) and ΔE_ν(12) for the particular νth band are intrinsically constant, disregarding the coordination and chemical environment of a given material, which follow the expressions:

where, ϕ_ν(r)(r ≠ r′) is the Bloch wave function at specific sites r. z = 0 and 12 represent an isolated atom and an atom in the ideal bulk, respectively. The exchange and overlap integrals are relative to the cohesive energy per bond 〈E_b〉. Any perturbation to β and γ causes shifts in the energy level, so the core level energy shifts depend on the bond energy.

With respect to ΔE_ν(12), the undercoordination induced core level shift ΔE_ν(z) for the surface follows:²²

With the given surface XPS spectra, we can determine the z-dependent νth energy of an isolated atom E_ν(0) and its BS (ΔE_ν(12)) by decomposing the spectra into the surface components and the bulk component with different coordination numbers with the relations derived from eqn (5):

Similarly, incorporating BOLS into the tight-binding theory yields the νth energy level shift of a nanocluster ΔE_ν(K) as follows:

ΔE_ν(K) = E_ν(K) − E_ν(0) = ΔE_ν(12) + ΔE_ν(12)Δ_H or E_ν(K) = E_ν(12) + ΔE_ν(12)Δ_H, (7)

where E_ν(K) is the XPS peak of the νth energy level of the nanocluster. If a cluster is approximately spherical, the relationship between its size and atomic number (N) is K⁻¹ = (3N/4π)^−1/3 ≈ 1.61N^−1/3,²³ thus, the N-dependence of the core-level BE E_ν(N) can be derived from eqn (3) and (7):

Generally, the size-induced BE shift for nanoclusters depends inversely on the size in the form of, E_ν(K) = A + BK⁻¹, where A and B are constants that can be determined by finding the intercept and the slope of the E_ν(K) line, respectively. According to the relationship between K and N, we can deduce the form of N versus BE shift into E_ν(N) = A + 1.61BN^−1/3 = A + B′N^−1/3. Comparison of the experimental scaling relationship with the theoretical expression yields,

Equating the BOLS theory prediction to the measurements gives the relation:This also allows us to determine the νth energy level of an isolated atom and its possible bulk shift:

Using eqn (6) or (11), we can readily determine the energy level E_ν(0) of an isolated atom and its BS ΔE_ν(12) by analyzing the measurements. Note that, for the same element, E_ν(0) and E_ν(12) are intrinsic constants for all surfaces, disregarding the coordination and chemical environment of a given material.

For detectable quantities that can be directly connected to bond identities such as bond nature, order, length and strength, we are able to predict the coordination number resolved local bond strain (ε(z)), the relative binding energy density (δE_D), the relative core-level binding energy shift (δE_z), and the relative cohesive energy per atom (δE_C) in the surface skins,

with z_ib = z_i/12. Subscript i denote the atomic layers counted from the outermost layer inwards. These basic magnitudes determine the related properties at the specific atomic sites. For instance, the atomic cohesive energy dominates the critical temperature of melting^24–26 and the binding energy density determines the elastic modulus.^27,28

2.2 DFT calculations

In order to verify the BOLS predictions on the coordination imperfection induced core-level entrapment and valence charge polarization, we conducted DFT calculations on the optimal bond relaxation, charge transfer,²⁹ and the energetic distribution of the core band and valence states of Mo nanoclusters with different structures,^30,31 as shown in Fig. 1. Moreover, we also compared the computational results with photoelectron spectroscopy measurements.

Fig. 1 Global minima of Mo_N clusters described by Finnis–Sinclair potentials.³⁰

DFT calculations were performed using the DMol3 code with a double numeric plus polarization basis set.³² For the DFT calculations, the potentials of the core electrons were assumed to be DFT semi-core pseudopotentials.³³ The DFT exchange–correlation potential utilized the local-density approximation (LDA), with the PWC function for geometry and electronic structures.^23,34 PWC is the default functional for DMol3 calculations. The self-consistency threshold of total energy was set at 10⁻⁶ au in the calculations. The tolerance limits for the energy, force, and displacements in the geometry optimization were considered at 10⁻⁵ Hartree, 0.002 Hartree Å⁻¹, and 0.005 Å, respectively.

3. Results and discussion

3.1 Surface core level quantum entrapment

To better understand the effects of coordination imperfection and the physical origin of surface core level shifts, we decomposed the Mo 3d_5/2 spectra of the clean Mo(100) and Mo(110) surfaces in Fig. 2.^12,35 Obeying the constraints given in eqn (5), the spectra from the clean surfaces of the Mo(100) and Mo(110) specimens were decomposed to three components, corresponding to the bulk (B) and surface skins S₂ and S₁ from higher (smaller absolute value) to lower BE after subtraction of the Tougaard background.³⁶ Table 1 summarizes the optimal component energies, the corresponding z_i, and the z-resolved local bond strain (ε_z), the relative binding energy density (δE_D), the relative core-level binding energy shift (δE_z), and the relative cohesive energy per atom (δE_C).

Fig. 2 Decomposition of the Mo 3d_5/2 spectra for (a) Mo(100)³⁵ and (b) Mo(110)¹² surfaces with three Gaussian components representing the bulk B, S₂, and S₁ components from higher (smaller absolute value) to lower BE, which obey the BOLS constraints given in eqn (5). The related derived information is listed in Table 1.

Table 1 Atomic-layer and crystal-orientation resolved effective CN (z), bond contraction (ε_z), relative BE shift (δE_z), relative atomic cohesive energy (δE_C), and the relative binding energy density (δE_D) determined from measured XPS profiles of Mo(100) and (110) surfaces under the established approach and criteria

	i	E3d5/2(i)	z	−εz	δEz (%)	−δEC (%)	δED (%)
Bulk	B	227.570	12.00	0	0	0	0
Mo(100)	S2	227.813	5.16	8.27	9.01	53.13	41.21
	S1	227.957	3.98	12.53	14.32	62.08	70.81
Mo(110)	S2	227.761	5.83	6.60	7.07	47.98	31.43
	S1	227.962	3.95	12.67	14.51	62.31	71.92

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From the decomposition, we obtained E_3d5/2(0), ΔE_3d5/2(z_i) and the standard deviation (σ) by using a least root-mean-square method. Fine-tuning of the CN values of the components will minimize σ and improve the accuracy of the effective CN for each sublayer. It was concluded that the 3d_5/2 BE of the bulk and surface skin shifts deeper from 2.707 to 3.100 eV with respect to that of isolated Mo (224.862 ± 0.004 eV) atoms. The following describes the obtained z-resolved 3d_5/2 BE shift for Mo skins:

This refinement leads to the effective atomic CNs of the top (second) Mo(100, 110) atomic layers as 3.98(5.16) and 3.95(5.83). With the derived z value and the 3d_5/2 BE for each XPS component, we could predict the z-resolved local bond strain ε_z(z), the relative BE shift δE_z(z), the relative atomic cohesive energy δE_C(z) and the relative binding energy density δE_D(z) of the Mo surface skins, as shown in Fig. 3. It was found that the undercoordination-induced local bond strain was contracted, the relative binding energy density was enhanced, the relative atomic cohesive energy was weakened and the relative core-level binding energy shifted positively up to 12.67%, 71.92%, 62.31%, 14.51%, respectively, for the three outermost atomic layers of the Mo surface. The fundamental information we have derived is of great importance in determining surface properties and processes.

Fig. 3 CN-resolved (a) local bond contraction ε_z(z) and relative BE shift δE_z(z); (b) relative atomic cohesive energy δE_C(z) and relative binding energy density δE_D(z).

3.2 Bond contraction, core level entrapment and valence charge polarization

According to BOLS-NEP, bonds between undercoordinated atoms in low-dimensional systems are shorter and stronger, which make the potential well deeper and hence, the core-level energy levels of these atoms drop correspondingly. Polarization induced by the densely trapped bonding electrons will take place if non-bonding electrons exist. To further confirm our predictions, we performed DFT calculations of the bond contraction, binding energy and valence density of states (DOS) of Mo_N clusters described by Finnis–Sinclair potentials³⁰ with different size and structure, as shown in Fig. 1. From the DFT results of the bond length and bond contraction between atoms in Mo_N clusters, we found that atoms in the cluster interior display no contraction equal to the ideal distance of the corresponding bulk atom, while the skin atoms with undercoordination numbers do contract, as listed in Table 2. The smaller the CN is, the further the bonds contract, that is, the extent of these clusters’ skin atom contraction depends on its CN. The results agree well with the reported bond contractions of Ag, Au, Cu, Ni, Pd, Pt, and Fe atomic chains.^37,38 We have also estimated the charge flow of these Mo nanoclusters using the Mulliken population analysis²⁹ and found that the electrons flow from the inner to the outermost layer of the clusters, which are also listed in Table 2. The negative sign represents charge loss. This indicates that the interaction between undercoordinated atoms induces quantum entrapment, and hence results in core electron entrapment from the local sites and the densification of charge and energy.

Table 2 DFT calculation results of the bond length (d_i), bond contraction coefficient (C_z) and charge transfer of Mo_N clusters. The charge transfer (Mulliken population analysis) of different structures reveals a charge flow from the inner to the outermost atomic shells due to quantum entrapment

Structure	d12(Å) (atom position 1–2)	Cz − 1 (%)	Charge transfer (e) (shell layer 1–2)
C2v15	2.612	−4.147	−1.028
C118	2.597	−4.697	−1.283
D320	2.574	−5.541	−1.333
C228	2.581	−5.284	−2.035
Cs47	2.493	−8.514	−4.137
C251	2.458	−9.798	−4.571
Oh59	2.560	−6.055	−5.952

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Fig. 4 shows the 4s BE (a) and LDOS (b) for the C_2v15, C₁18, D_s20, C₂28, C_s47, C₂51 and O_h59 structures calculated by DFT. It is readily shown that the 4s level shifts from 61.229 eV for the Mo₅₉ nanocluster to 61.620 eV for the Mo₁₅ nanocluster, while the LDOS peak shifts from −0.897 eV for Mo₅₉ to 0.160 eV for Mo₁₅. This illustrates that both core-level atom localization-entrapment and valence charge polarization take place in these Mo nanoclusters, and hence lead to positive core level shifting and negative valence band shifting. Both of them exhibit stronger bonds as the size decreases. These DFT calculations further confirm our BOLS-NEP predictions.

Fig. 4 Undercoordination induced (a) core-level energy level entrapment and (b) valence band polarization of Mo clusters (C_2v15, C₁18, D_s20, C₂28, C_s47, C₂51, O_h59), illustrating that the extent of entrapment and polarization depends on the cluster size.

3.3 N-resolved E_ν(N)

Choosing Mo₅₉ cluster as the standard reference, we can obtain its dimensional size K = R/d_b = 1.714, for which R is 4.672 Å and the d_b is 2.725 Å. According to eqn (1), we can estimate z₁ = 4(1 − 0.75/K) = 2.25 and z₂ = z₁ + 2 = 4.25 and thus obtain C₁ and C₂ being 0.7356 and 0.8865, respectively. Fig. 5 shows that the E_4s(N) of Mo nanoclusters depends linearly on N^−1/3, which gives the y-intercept and the slope as 60.760 and 2.062, respectively. According to the y-intercept and slope derived from N-induced Mo E_4s(N) and eqn (11), we can obtain:

E_4s(12) = 60.760 eV and ΔE_4s(12) = 3.389/τ

Fig. 5 The N-resolved E_4s of free Mo_N nanoclusters.

Because the shape of the Mo nanoclusters is not an ideal sphere, the shape factor τ is uncertain, which should be greater than the unit and smaller than three. Although the bulk shift of Mo ΔE_4s(12) is absent, we have obtained ΔE_3d5/2(12) = 2.707 eV. Though the shifts of different core level energies have a slight deviation, the core level band of the nanoclusters shifts down as a whole, so we have assumed that the BS for E_4s and E_3d5/2 shifts the same way to obtain the information for E_4s. Therefore, we can determine the value of τ as 1.252. Thus, N-resolved E_4s(N) follows:

Such a consistency between DFT calculations and BOLS predictions evidences that local bond contraction and quantum entrapment lead to a densification of the mass, charge and energy near the undercoordinated atomic sites, which result in the enhancement of the Hamiltonian and hence, a positive BE shifting for Mo skins and nanoclusters. Therefore, the interaction between undercoordinated atoms dominates the behavior of BE shifting.

4. Conclusions

Consistency between DFT calculations and photoelectron spectroscopy measurements confirmed our BOLS-NEP predictions of the atomic CN effects on local bond relaxation, electron binding-energy shifting, and valence charge transfer of Mo skins and nanoclusters. It was clarified that Mo 3d_5/2 (4s) shifts positively by 2.707 eV from a value of 224.862 (58.053) eV for an isolated atom to 227.569 (60.760) eV for the bulk. The interaction between under-coordinated atoms caused local strain, local densification and entrapment of the core electrons, as well as valence charge polarization, which perturb the Hamiltonian and hence dominate the unusual behaviour of Mo surfaces and nanoclusters. These findings will be useful in the application of Mo skins and nanoclusters in practical situations, such as catalytic enhancement and applications in electronics and optics.

Acknowledgements

We gratefully acknowledge financial support from the NSF (Grants no. 11172254 and no. 11402086).

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