1168
2009, 113, 1168–1170 Published on Web 01/08/2009
Surface Energy and Electronic Structures of Ag Quasicrystal Clusters D. Liu, J. S. Lian, and Q. Jiang* Key Laboratory of Automobile Materials (Jilin UniVersity), Ministry of Education, and Department of Material Science and Engineering, Jilin UniVersity, Changchun 130022, China ReceiVed: NoVember 20, 2008; ReVised Manuscript ReceiVed: December 24, 2008
Size-dependent surface energy associated with atomic number N, γ(N), of Ag quasicrystal clusters was investigated using density functional theory and broken-bond theory. It is found that the value of γ(N) is 0.55-0.66 eV/atom, similar to the value of their counterparts in bulk. This is due to the increase of cohesive energy Ec(N) and the decrease of surface coordinate number associated with the decreasing of N. As N decreases, the partial density of states shifts to higher binding energy, and there is electronic charge transfer from the sp band into the d band, which leads to the increase of cohesive energy. 1. Introduction
2. Simulation Details and Theoretical Model
In the past decade, atomic clusters with large surface-tovolume ratios have been intensively studied due to their unique chemical and physical properties compared with those of their counterpart bulk crystals.1,2 One of the applications of these materials is to act as a catalyst by utilizing the surface energy γ(N), where N is the number of atoms in a cluster.3,4
The first-principles density functional theory (DFT)10,11 was performed using DMOL code,12,13 in which each electronic wave function was expanded in a localized atom-centered basis set. Each basis function was defined numerically on a dense radial grid, using a state-of-the-art delocalized internal coordinate optimization scheme. The quality Fine of geometry optimization was chosen, where the convergence tolerance of energy was 1.0 × 10-5 Ha (1 Ha ) 27.2114 eV), the maximum force was 0.002 Ha/Å, and the maximum displacement was 0.005 Å. The local density approximation (LDA) with Perdew-Wang parametrization14 and Ceperly and Alder15 (PWC) functions was employed as the exchange-correlation value. DFT semicore pseudopots (DSPP)16 developed were included via a semilocal pseudopotential for all-electrons calculations, where the effect of core electrons was substituted by a simple potential including some degree of relativistic effects. In addition, double numerical plus polarization (DNP)10 was chosen as the basis set with an orbital cutoff of 4.4 Å. To achieve the convergent Ec(N) values of Ag clusters, we used smearing techniques17 with an appropriate value of 0.005 Ha. Note that in our simulations, without using the smearing technique, Ec(N) is only convergent for the IH13 structure, where the difference of Ec(13) values with or without smearing of 0.005 Ha is 2%. For other structures, the difference between Ec(N) with smearing of 0.005 Ha and that with smearing of 0.001 Ha is only 0.5%. Thus, smearing of 0.005 Ha is taken without big error. In all cases, spin polarization was introduced. Ec(∞) ) -3.51 eV/atom, and the nearestneighbor distance was h(∞) ) 2.85Å through optimization of a standard crystalline cell of Ag. For the PDOS calculation, the empty band was chosen as 12. To calculate γ(∞) values, 3 × 3 × 7 slabs of (111) and (100) planes were separately established. A vacuum region being larger than 12 Å was set to separate adjacent slabs. The K-point was set as 3 × 3 × 1 which is fine for the slabs. The γ(∞) could be calculated as18
It is known that the bulk surface energy γ(∞) for Ag is about 1.2-1.4 J/m2,5 while γ(N) ) 7.2 J/m2 for Ag nanocrystals, which was determined by detecting the size-dependent evaporation temperature.6 In contrast, from a thermodynamic viewpoint with the size-dependent cohesive energy Ec(N), it is found that γ(N) decreases with N.7 When the size of Ag clusters falls into a range of 0.5-5.5 nm, γ(N) ) 1-2.2 J/m2 ≈ γ(∞).8 Therefore, it is necessary to clarify the correlation between the size and surface energy of Ag on the nanometer scale. Clusters with tens or hundreds of atoms usually have quasicrystalline structures such as the icosahedron (IH), the truncated decahedron (DH), and the truncated octahedron (TO) with a large proportion of atoms located at edges and vertices. In general, these atoms have a lower coordinate number (CN) than those at facets.9 Thus, the average CN of surface atoms Zs(N) decreases with N, leading to an increase of γ(N). It is noted that the cluster surface is not flat.9 Considering Ec(N) > Ec(∞) and γ(N) ∝ Ec(N), where Ec(∞) is the bulk cohesive energy and Ec has a unit of eV/atom, the decline of γ(N) is expected. Since the two factors result in different changes in direction of γ(N), which are not quantitatively determined, the γ(N) function remains unclear. In this letter, we report our calculation of γ(13 < N < 147) values of Ag clusters obtained by DFT with consideration of the broken bond model. The partial density of states (PDOS) and Mulliken charge transfer of the clusters are also investigated to understand the variation of electron structure. * To whom correspondence should be addressed. Fax: +86 431 85095876. E-mail:
[email protected].
10.1021/jp810220f CCC: $40.75
γ(∞) ) (Eslab - 7EBulk) ⁄ 2
(1)
where Eslab and EBulk are separately the total energies per unit cell of the seven-layer slab and the bulk crystal. 2009 American Chemical Society
Letters
J. Phys. Chem. C, Vol. 113, No. 4, 2009 1169 TABLE 1: Ns, Ec(N), and Zs(N) Values Determined by DFT-LDA Simulationa γ(N) γ(N) clusters N Ns -Ec(N) Zs(N) (eV/atom) (J/m2) -Ecs(N) -Eci(N) IH13 13 12 TO38 38 32 IH55 55 42 DH75 75 57 TO79 79 60 DH101 101 72 IH147 147 92
Figure 1. Atomic number dependence of Ag cluster structures.
Figure 1 shows structures of Ag clusters with different N values. As a spherical structure, IH may be energetically favorable.9 However, the multiply twinned IH with a high symmetry could not be perfectly packed at some N values, which led to stabilization of other structures in FCC metallic clusters at N < 150,19,20 where the TO structure at N ) 38 and 79 and DH structure at N ) 75 and 101 are also energetically favorable. Thus, TO and DH are also considered. The chosen clusters are all with the magic numbers of atoms,19,20 namely, N ) 13, 55, and 147 for IH, N ) 38 and 79 for TO, and N ) 75 and 101 for DH. Let Z(∞) denote the bulk CN; considering the surface bond relaxation, isothermal γ(∞) could be determined as18
γ(∞) ) -Ec(∞){1 - [Zs(∞) ⁄ Z(∞)]1⁄2}
(2)
where Zs(∞) denotes the CN of the bulk surface. Although eq 2 underestimates γ(∞) values compared with experimental results,5 it is quite in agreement with the simulation results18,21 since LDA simulation underestimates γ(∞) values too due to the electronic surface error.22,23 To guarantee the correspondence between the theory in light of eq 2 and the simulation, eq 2 is taken. Extending eq 2 to the cluster in nanometer scale, there is
γ(N) ) -Ec(N){1 - [Zs(N) ⁄ Z(∞)]1⁄2}
(3)
The γ(N) could also be regarded as the difference energies between the cohesive energy of surface atoms Ecs(N) and that of interior atoms Eci(N), namely, γ(N) ) Ecs(N) - Eci(N). Considering the additional relationship of Ec(N) ) (1 - Ns/ N)Eci(N) + (Ns/N)Ecs(N), where Ns is the number of the surface atoms, and substituting the above two equations into eq 3, we have
Ecs(N) ) Ec(N){1 + (Ns ⁄ N - 1){1 - [Zs(N) ⁄ Z(∞)]1⁄2}} (4.1) Eci(N) ) Ec(N){1 + (Ns ⁄ N){1 - [Zs(N) ⁄ Z(∞)]1⁄2}} (4.2) 3. Results and Discussion The simulated values of Ec(N) and Zs(N) are shown in Table 1. Ec(N) > Ec(∞), while Ec(N) increases with decreasing N. In contrast, Zs(N) < Zs(∞) with Zs111(∞) ) 9 as the (111) facet has the lowest γ(∞) for FCC metals where the subscript denotes the index of the plane. Zs(N) approximately decreases with decreasing N. Zs(13) ) 6 where all surface atoms are located at the vertices sites. On the basis of the Ec(N) and Zs(N) values, γ(N) values calculated with eq 3 are listed in Table 1. The γ(N) ) 0.55-0.66 eV/atom because the decrease of Zs(N) is compensated by the increase of Ec(N), which is almost independent of N. For comparison, γ111(∞) ) 0.55 eV/atom and γ100(∞) ) 0.69 eV/ atom are separately calculated based on DFT-LDA simulation, similar to the calculated results of eq 2 of γ111(∞) ) 0.47 eV/
2.13 2.64 2.79 2.82 2.83 2.90 2.99
6 6.75 7.43 7.4 7.4 7.53 7.96
0.62 0.66 0.59 0.61 0.61 0.60 0.55
0.84 1.17 1.07 1.23 1.25 1.25 1.14
2.09 2.54 2.65 2.67 2.68 2.73 2.79
2.71 3.20 3.24 3.28 3.29 3.33 3.34
a The γ(N) are in units of eV/atom and J/m2; Ecs(N) and Eci(N) are calculated with eqs 3, 4.1, and 4.2.
atom and γ100(∞) ) 0.64 eV/atom. Thus, γ111(∞) < γ(N) < γ100(∞). The γ(N) in units of J/m2 is also calculated. For bulk (111) and (100) planes, the areas occupied by per surface atom are separately A111 ) (3)1/2h2(∞)/2 and A100 ) h2(∞).5 Through dividing, these unit surface areas are γ111(∞) ) 1.2 J/m2 and γ100(∞) ) 1.36 J/m2. Since Ag clusters could be a sphere in shape, the area occupied by per surface atom could be approximately deduced as As(N) ) πD2(N)/Ns.8 D(N) is the cluster diameter as D(N) ≈ (N)1/3h(N) ≈ (N)1/3h(∞) with neglect of the small bond’s contraction.24 Through dividing As(N), γ(N) in units of J/m2 is obtained, which is also shown in Table 1. It is found that γ(13) ) 0.84 J/m2, which is much smaller than the other γ(N) values. This is mainly due to the smaller Zs(13) ) 6, which increases As(N). Ecs(N) and Eci(N) values determined by eq 4 are presented in Table 1. Ecs(N) > Ec(N) > Eci(N) and Eci(∞) ≈ Ec(∞). Ecs(N) gradually increases with N from Ecs(147) ) -2.79 eV/atom to Ecs(13) ) -2.09 eV/atom. Since Ecs(∞) ) Eci(∞) + γ(∞), Ecs111(∞) ) -2.96 eV/atom and Ecs100(∞) ) -2.82 eV/atom. Thus, Ecs(N) > Ecs100(∞) > Ecs111(∞). In addition, Eci(N) > Eci(∞), and Eci(N) has the same tendency as Ecs(N) with N from Eci(147) ) -3.34 eV/atom to Eci(13) ) -2.71 eV/atom. Increases of both Ecs(N) and Eci(N) result in the approximately constant γ(N) value, while γ(N) ) Ecs(N) - Eci(N). For nanocrystals, it is known that Eci(N)/Ecs(N) ≈ σ2ms(N)/ σ2mi(N) ) R, where σ2ms(N) and σ2mi(N) are separately the mean square displacements of surface and interior atoms when R is taken as a constant.25 The above equation has been widely used to illustrate many size-dependent phenomena of nanocrystals.25 In our calculations, R ) 1.2-1.3, while R111 ) 1.2 and R100 ) 1.25. Thus, the Eci(N), Ecs(N), and γ(N) values obtained in our work are applicable for nanocrystals. The PDOS of the surface atoms and the interior atoms in units of per atom are shown in Figures 2 and 3, respectively. In Figure 2, the valence band of the Ag(100) plane is enhanced compared with that of the Ag(111) plane shown in Figure 2h and i, which confirms the result of Ecs100(∞) > Ecs111(∞). For IH-147 and DH-101 in Figure 2f and g, the valence bands show a slight red shift, which results in Ecs(101) > Ecs(147) > Ecs100. As N decreases, the red shift becomes more evident with the energetic series of Ecs(79) < Ecs(75) < Ecs(55) < Ecs(38) < Ecs(13), as shown in Table 1. As N further reduces, multipeaks or discontinuous valence bands appear. In Figure 3d-g, Eci(75 e N e 147) ) -3.28 to -3.34 eV/atom; the valence bands remain the bulk location, although the intensity of the higher energy peaks is enhanced compared with that of the bulk in Figure 3h. When N ) 55 in Figure 3c, a red shift and multipeaks are observed, and an extra peak emerges at the lower end of the energy. When N ) 13, the valence bands are strongly expanded at both ends and depleted in the middle (Figure 3a).
1170 J. Phys. Chem. C, Vol. 113, No. 4, 2009
Letters with those of bulk, which corresponds to previous conclusions28,29 where electronic charges transfer from the sp bands into the d band, which weakens the s-p-d hybridization30 and induces the Ecs(N) and Eci(N) increase. 4. Conclusions
Figure 2. PDOS of surface atoms of Ag clusters at different N values. The PDOS of their counterparts in bulk is given for comparison.
In summary, γ(N) values of Ag quasicrystalline clusters have been investigated. They are about 0.55-0.66 eV/atom, which is between γ111(∞) and γ100(∞). This phenonmenon results from the common effects of Ec(N) increasing and Zs(N) decreasing with decreasing N while both Ecs(N) and Eci(N) increase with a red shift of the corresponding valence bands. The Mulliken analysis shows that there is sp f d charge transfer for both the surface and interior atoms, which further demonstrates the increase of Ecs(N) and Eci(N). Acknowledgment. The authors acknowledge the financial supports from National Key Basic Research and Development Program of China (Grant No. 2004CB619301) and “985 Project” of Jilin University. References and Notes
Figure 3. PDOS of interior atoms of Ag clusters at different N values. The PDOS of their counterparts in bulk is given for comparison.
TABLE 2: Charge Transfer (CT) of the Surface and Interior Atoms of Ag Clustersa surface clusters IH13 TO38 IH55 DH75 TO79 DH101 IH147 bulk (100) bulk (111)
interior
4d
5s
5p
CT
4d
5s
5p
CT
9.828 9.797 9.79 9.777 9.774 9.774 9.777 9.772 9.75
0.886 0.881 0.895 0.884 0.883 0.882 0.885 0.868 0.874
0.3 0.433 0.339 0.358 0.361 0.362 0.357 0.372 0.398
-0.014 -0.011 -0.024 -0.019 -0.018 -0.018 -0.019 -0.012 -0.022
9.796 9.736 9.721 9.723 9.726 9.714 9.706 9.704
0.478 0.69 0.662 0.709 0.709 0.736 0.734 0.804
0.558 0.504 0.538 0.506 0.509 0.509 0.525 0.492
0.168 0.07 0.079 0.062 0.056 0.041 0.035 0
a The counts of electrons in 4d, 5s, and 5p orbitals are also shown.
Table 2 shows the Mulliken charge transfers of the surface and interior atoms. It is found that for all clusters, the surface is negatively charged, and the interior is positively charged, which is in agreement with the previous simulation results for Au-Ag clusters.26 Bonding of transitional metals is mainly due to the overlap among d shells accompanied with the hybridization of s-p-d electrons.27 For bulk Ag, due to this s-p-d hybridization, some electrons in 4d and 5s orbitals transit to the 5p one, which results in the electronic configuration of 4d105s1 transforming to 4d9.7045s0.8045p0.492. For clusters, there is a loss of 5s or 5p electrons and gain of 4d electrons compared
(1) Halperin, W. P. ReV. Mod. Phys. 1986, 58, 553. (2) Ferrando, R.; Jellinek, J.; Johnston, R. L. Chem. ReV. 2008, 108, 846. (3) Turner, M.; Golovko, V. B.; Vaughan, O. P. H.; Abdulkin, P.; Berenguer-Murcia, A.; Tikhov, M. S.; Johnson, B. F. G.; Lambert, R. M. Nature 2008, 454, 981. (4) Coquet, R.; Howard, K. L.; Willock, D. J. Chem. Soc. ReV. 2008, 37, 2046. (5) Jiang, Q.; Lu, H. M.; Zhao, M. J. Phys.: Condens. Matter 2004, 16, 521. (6) Nanda, K. K.; Maisels, A.; Kruis, F. E.; Fissan, H.; Stappert, S. Phys. ReV. Lett. 2003, 91, 106102. (7) Lu, H. M.; Jiang, Q. J. Phys. Chem. B 2004, 108, 5617. (8) Medasani, B.; Park, Y. H.; Vasiliev, I. Phys. ReV. B 2007, 75, 235436. (9) Baletto, F.; Ferrando, R.; Fortunelli, A.; Montalenti, F.; Mottet, C. J. Chem. Phys. 2002, 116, 3856. (10) Hohenberg, P.; Kohn, W. Phys. ReV. 1964, 136, B864. (11) Kohn, W.; Sham, L. J. Phys. ReV. 1965, 140, A1133. (12) Delley, B. J. Chem. Phys. 1990, 92, 508. (13) Delley, B. J. Chem. Phys. 2000, 113, 7756. (14) Perdew, J. P.; Wang, Y. Phys. ReV. B 1992, 45, 13244. (15) Ceperley, D. M.; Alder, B. J. Phys. ReV. Lett. 1980, 45, 566. (16) Delley, B. Phys. ReV. B 2002, 66, 155125. (17) Seminario, J. M.; Politzer, P. Theoretical and Computational Chemistry; Elsevier Science: Amsterdam, The Netherlands, 1995. (18) Methfessel, M.; Hennig, D.; Scheffler, M. Phys. ReV. B 1992, 46, 4816. (19) Doye, J. P. K.; Wales, D. J. New J. Chem. 1998, 22, 733. (20) Grigoryan, V. G.; Alamanova, D.; Springborg, M. Phys. ReV. B 2006, 73, 115415. (21) Swart, J. C. W.; van Helden, P.; van Steen, E. J. Phys. Chem. C 2007, 111, 4998. (22) Mattsson, A. E.; Wixom, R. R.; Armiento, R. Phys. ReV. B 2008, 77, 155211. (23) Carling, K.; Wahnstro¨m, G.; Mattsson, T. R.; Mattsson, A. E.; Sandberg, N.; Grimvall, G. Phys. ReV. Lett. 2000, 85, 3862. (24) Jiang, Q.; Liang, L. H.; Zhao, D. S. J. Phys. Chem. B 2001, 105, 6275. (25) Jiang, Q.; Zhang, Z.; Li, J. C. Chem. Phys. Lett. 2000, 322, 549. (26) Chen, F.; Johnston, R. L. Acta Mater. 2008, 56, 2374. (27) Gelatt, C. D.; Ehrenreich, H.; Watson, R. E. Phys. ReV. B 1977, 15, 1613. (28) Tersoff, J.; Falicov, L. M. Phys. ReV. B 1981, 24, 754. (29) Liu, Z. P.; Hu, P.; Alavi, A. J. Am. Chem. Soc. 2002, 124, 14770. (30) Cook, S. C.; Padmos, J. D.; Zhang, P. J. Chem. Phys. 2008, 128, 154705.
JP810220F
