Vacancy Formation Energy and Cohesive Energy of

their counterpart in bulk [38]. One of the applications of these materials is to act as catalysts with certain Ec(N ) and γsv(N ) due to huge surface-volume ratio, where N is the number of atoms in a cluster [39]. If a metallic cluster has a spherical shape, the corresponding N value is named magic number. These kinds of clusters are easy to study where some rules could be found, which will be considered in this section. The best known structures of this kind of structure or cluster with magic number are quasi-crystalline structures such as icosahedron (IH), truncated decahedron (DH), and truncated octahedron (TO) with large proportion of atoms located at edges and vertices [40].

3.5 Cohesive Energy 109

The basic understanding of energetic term of clusters is the cohesive en-ergy of atoms at surface and interior, Ecs(N ) and Eci(N ), which are the mean values of all surface and interior atoms of clusters. Both can be determined by computer simulation techniques. As a result, Ec(N ) and γsv(N ) are ob-tained. More exactly, the cohesive energy of atoms at a special x site, E_cx(N ), is more complicated and in some cases is more important than Ec(N ) one.

This is because many properties, such as adsorption, catalysis and optical behavior of clusters, are exhibited by atoms at special surface locations of clusters [41]. Moreover, alloy clusters show superior performance compared with a single element cluster in catalytic and optical properties [42]. To de-termine the degree of segregation or mixing in an alloy cluster, besides the surface sites, E_cx(N ) of the interior sites also needs to be understood.

E_cx(N ) values cannot be directly obtained by using simulation tech-niques. The vacancy formation energy of the x site E_vx(N ) in a cluster however could be determined using simulation techniques [43], which can be related to E_cx(N ). For bulk crystals, the vacancy formation energy Ev(∞) is approximately a fraction of Ec(∞), such as E^v(∞)/E^c(∞) ≈ −0.3 for the transitional metals [44]. For nanocrystals, a thermodynamic relation-ship has been deduced as Ev(N )/Ec(N ) = Ev(∞)/E^c(∞) where E^v(N ) is the average value of E_vx(N ) [45]. Since clusters usually have non-crystalline and quasi-crystalline structures, this linear relationship may be invalid and E_cx(N )∼ Evx(N ) relationship still needs to be clarified.

In the following, Ag clusters with the magic number are introduced as examples to illustrate the situations of clusters and the corresponding Ec(N ) and Ev(N ) where the typical IH (N = 13, 55 and 147), DH (N = 75 and 101) and TO (N = 38) structures are taken as shown in Fig. 3.13. The specific surface and interior sites are marked with numbers.

Ev(∞) denotes the lowest energy to remove an atom from a selected site, and usually the atom is brought to an assumed reservoir which determines the atomic chemical potential [46]. For single element crystals, this potential is Ec(∞). The physical meaning behind this is that the removed atom is brought to a kink site at surface [46]. Although the cohesive energy of the under-coordinated kink site Eck(∞) is greater than Ec(∞), it has been widely reported that the binding energy of the atoms at the kink site Ebk(∞) is just equal to−E^c(∞) [47]. This is mainly because when an atom is bound at the kink site, besides its cohesive energy turning from 0 to Eck(∞), the cohesive energy of the coordinated atoms also decreases since these atoms all gain one extra bond. Thus, in Density Functional Theory (DFT) simulation, Ev(∞) has been deduced as [46]

Ev(∞) = E(1) + E(∞, N − 1, 1) − E(∞, N, 0) + Ec(∞) (3.91) where E(1) is the total energy of a single atom, E(∞, N, 0) and E(∞, N −1, 1) are separately the total energy of the super cell and that after the atom at x site is removed. Since clusters are quite different from bulk crystals in both structure and energy, a new reservoir should be assumed to calculate E_vx(N ).

110 Chapter 3 Heat Capacity, Entropy, and Nanothermodynamics

Fig. 3.13 Ag cluster structures with different magic numbers N. The different surface and interior sites are marked with numbers and the corresponding site names are shown in Table 3.5.

With the same consideration as bulk crystals, it is simply assumed that the new reservoir is determined by Ec(N ) where Ebk(N ) =−Ec(N ).

Table 3.5 Ec(N ) and Ecx(N ) in eV.atom⁻¹, Evx(N ) in eV, N_x and Z_x(N ) of Ag clusters obtained from DFT simulation and Eqs. (3.92) and (3.95)

Cluster −E^c(N ) Atomic sites N_x Z_x(N ) Evx(N ) −E^cx(N )

IH13 2.13 (1) vertex 12 6 0.44 2.10

(2) core 1 12 0.79 2.44

TO38 2.64 (1) (111) facet 8 9 1.21 3.10

(2) vertex 24 6 0.5 2.39

(3) sub-layer 6 12 1.16 3.05

IH55 2.79 (1) edge 30 8 1.05 2.77

(2) vertex 12 6 0.84 2.56

(3) sub-layer 12 12 1.33 3.05

(4) core 1 12 1.15 2.87

DH75 2.82 (1) (111) facet 10 9 1.15 3.05

(2) top edge 10 8 0.9 2.80

(3) notch edge 5 10 1.35 3.25

(4) top vertex 2 6 0.4 2.3

(5) notch vertex 10 7 0.84 2.74

(6) lateral vertex 20 6 0.5 2.4

(7) sub-top vertex 2 12 1.18 3.08

(8) sub-top edge 10 12 1.36 3.24

(9) sub-(111) facet 5 12 1.04 2.94

(10) core 1 12 1.32 3.22

3.5 Cohesive Energy 111 Continue Cluster −E^c(N ) Atomic sites N_x Z_x(N ) Evx(N ) −E^cx(N )

DH101 2.9 (1) (111) facet 10 9 1.04 3.08

(2) top edge 10 8 0.82 2.86

(3) notch edge 10 10 1.24 3.28

(4) lateral edge 10 7 0.61 2.65

(5) top vertex 2 6 0.32 2.36

(6) notch vertex 10 7 0.65 2.69

(7) lateral vertex 20 6 0.37 2.41

(8) sub-notch edge 10 12 1.43 3.47 (9) sub-top vertex 10 12 1.06 3.10

(10) sub-top edge 2 12 1.16 3.20

(11) sub-(111) facet 5 12 1.16 3.20

(12) core 2 12 1.31 3.35

IH147 2.99 (1) (111) facet 20 9 1.14 3.18

(2) edge 60 8 0.81 2.85

(3) vertex 12 6 0.3 2.34

(4) sub-edge 30 12 1.25 3.29

(5) sub-vertex 12 12 0.99 3.03

(6) second sub-layer 12 12 1.28 3.32

(7) core 1 12 0.42 2.46

To confirm the validity of the above relationship, more detailed discussion is given. At bulk surfaces, CN of the atoms at the close packed (111) facet and at the kink site are separately Zs(111)= 9 and Zsk= 6 [48]. Hence, three neighboring bonds of an atom should be broken when making a kink at the plane. For clusters, the average CN of the atoms at the close packed sur-face Zs(N ) = 6− 8 [49], which brings out the CN of corresponding kink sites Zsk(N ) being 3− 5. For instance, for Ag IH13 and DH75 clusters, Ec(13)/Ec(∞) = 0.61 and E^c(75)/Ec(∞) = 0.8 with Z^s(13) = 6 and Zs(75)

= 7.4 [49]. Thus, Zsk(13)≈ 3 and Z^sk(75)≈ 4. It has been reported that at Cu bulk surfaces, the binding energy of the Cu adatoms at the hollow sites of the (111) and (100) facets keeps the ratios of Eb(111)(∞)/Ec(∞) = −0.64 and Eb(100)(∞)/Ec(∞) = −0.8 [47]. Since the CN of these sites are separately Zs(111)(∞) = 3 and Zs(100)(∞) = 4 while the crystalline structure of Ag is also fcc, these sites should be similar to the kink sites of Ag IH13 and DH75 clusters. Thus, Ebk(13)/Ec(∞) ≈ −0.64 and Ebk(75)/Ec(∞) ≈ −0.80 are approximately valid and the assumption of Ebk(N )≈ −Ec(N ) is reasonable.

With the above consideration for the new reservoir, E_vx(N ) is defined as E_vx(N ) = E(1) + E_x(N− 1, 1) − E(N, 0) + Ec(N ) (3.92) where E(N, 0) and E_x(N− 1, 1) are the total energy of the cluster with N atoms and that after the atom at x site is removed. For transitional metals, it has been found that Ev(∞)−Ev(111)(∞) ≈ Ecb(∞)−Ec(111)(∞) = γ(111)(∞) where γ is the surface energy, which says that the difference of Ev between the bulk surface and interior is essentially induced by the corresponding dis-crepancy in Ec. Following the same consideration, the difference of E_vx(N ) in the same cluster is assumed to be mainly decided by the discrepancy in

112 Chapter 3 Heat Capacity, Entropy, and Nanothermodynamics

E_cx(N ), there is

E_vy(N )− Evx(N )≈ Ecy(N )− Ecx(N ) (3.93) where subscript “y” denotes another site in the cluster, which differs from

“x”. Moreover, E_cx(N ) could also be deduced as E_cx(N ) = [N Ec(N )−

z

N_zE_cz(N )]/N_x (3.94)

where N_x and N_z are the numbers of atoms of the x and z sites. Inserting Eq. (3.94) into Eq. (3.93), it reads

E_cx(N ) = Ec(N )− {

y

N_y[E_vy(N )− Evx(N )]}/N. (3.95)

In Eq. (3.95), Ec(N ) values could be obtained by simulations, E_vy(N ) and E_vx(N ) values are calculated in terms of Eq. (3.92) and simulation results.

Moreover, γ(N ) is defined as [49]

γ(N ) = Ecs(N )− Eci(N ) (3.96) where Ecs(N ) and Eci(N ) are separately the average cohesive energy of the surface and interior atoms of the considered clusters.

Based on Eq. (3.91), Ev(∞) = 1.18 eV, which agrees with experimental and simulation results of 1.11 – 1.24 eV for bulk Ag crystals [45]. The calcu-lated E_vx(N ) values of Ag clusters in light of Eq. (3.92) are shown in Table 3.5. The vertex sites have comparatively small E_vx(N ) values as Ev-vertex(N )

= 0.3 – 0.84 eV, or the sites are easy to form vacancy compared with other surface sites. This result is not difficult to understand since Zvertex(N ) = 6 except at the notch vertex sites of DH structures. In addition, E_vx(N ) firstly increases with increasing N until a maximum appears at Ev-^vertex(55) = 0.84 eV and then it decreases. Namely, vacancy is easier to form at the vertex sites with increasing N when N > 55.

Clusters would transform to nanocrystals as N further increases. Al-though nanocrystals usually keep the TO structure thermodynamically, the IH and DH structures could also be metastable due to some kinetic factors [50]. Simulation results show that Au nanocrystals in a size range of 3–8 nm have the “Chui icosahedron” (c-IH) structure [51], which is truncated from the IH structure where all atoms at the vertex site are removed. A thermo-dynamic model suggests that there is a critical size beyond which the c-IH structure is energetically more favorable than the IH structure [52]. This con-clusion corresponds to the result that clusters or nanocrystals with larger size tend to form vacancies at the vertex sites.

For the atoms at (111) facets [Z(111)(N ) = 9] and edge [Zedge(N ) = 8]

sites, Ev(111)(N ) = 1.04 – 1.21 eV and Ev-^edge(N ) = 0.81 – 1.05 eV, which are much larger than those of the bulk (111) and (100) facets of Ev-⁽¹¹¹⁾(∞)

3.5 Cohesive Energy 113

= 0.77 eV and Ev-⁽¹⁰⁰⁾(∞) = 0.53 eV. Hence, vacancies are more difficult to form at these sites for clusters than for the corresponding bulks. For the interior sites of Ag clusters, most Ev-^interior(N ) values are in the range of 1 – 1.43 eV, which is approximately equal to or even larger than Ev(∞) = 1.18 eV. Accordingly, vacancies are more difficult to form at some interior sites of clusters than those at the interior of bulk crystals. The exceptions are the core sites of IH13 and IH147 with Ev-^core (13) = 0.79 eV and Ev-^core (147)

= 0.42 eV. Therefore, the core site of the IH structure may be unoccupied.

On the other side, the calculated E_cx(N ) values in terms of Eq. (3.94) are also shown in Table 3.5. For the surface sites with Z_sx(N ) = 6, Ec-^vertex(N ) values are in the range of (−2.3) − (−2.56) eV with only exception of Ec-^vertex(13) =−2.10 eV. Ec-^vertex(N ) does not show obvious size dependence like Ev-vertex(N ). Similarly, E_cx(N ) keeps almost the same with the same Z_sx(N ) when Z_sx(N ) = 7 − 10. For the bulk surface, Ecs(∞) ≈ [Zs(∞)/Z(∞)]^1/2Ec(∞) has been demonstrated using simulation techniques [53]. E_cx(N )∼ Zx(N ) relationship is similar to Ecs(∞) ∼ Ec(∞) one. Thus, for the most surface sites of clusters, E_cx(N ) ≈ Ecs(∞) when Zsx(N ) = Zs(∞). As mentioned above, even if Ecx(N ) ≈ Ecs(∞), surface atoms of clusters could not easily be separated to form vacancy. This is because for forming a vacancy at the cluster surface, chemical potential of the assumed reservoir has been changed from Ec(∞) to Ec(N ), which increases difficulty of the vacancy formation.

For most interior atoms of clusters, E_cix(N ) = (−2.87)− (−3.35) eV with Z_ix(N ) = 12. Although E_cix(N ) > Ec(∞), some interior sites are harder to form vacancy than bulk due to the change of the chemical potential of the reservoir from Ec(∞) to Ec(N ). The exceptions are the core sites of IH13 and IH147 with Ec-^core (13) = 2.44 eV and Ec-^core (147) = 2.46 eV where there are smaller vacancy formation energy of Ev-^core (13) = 0.79 eV and Ev-^core (147) = 0.42 eV.

Through averaging the E_csx(N ) and E_cix(N ) values, Ecs(N ), Eci(N ) and γ(N ) values in terms of Eq. (3.96) are calculated and shown in Table 3.6.

Both Ecs(N ) and Eci(N ) increase as N decreases. γ(N ) values are in the range of 0.46 – 1.06 J·m⁻², which is smaller than the corresponding bulk value γ(N ) = 1.2 J·m⁻² [53]. Since the surface atoms are mostly located at the edge and vertex sites, they should have more broken bonds and higher Table 3.6 Ecs(N ), Eci(N ), γ(N ) and Zs(N ) of Ag clusters obtained from DFT simulation and Eqs. (3.95) and (3.96)

Clusters Zs(N ) γsv(N ) /(eV·atom⁻¹)

γsv(N )

/(J·m⁻²) −Ecs(N ) −Eci(N )

IH13 6 0.34 0.46 2.1 2.44

TO38 6.75 0.48 0.85 2.57 3.05

IH55 7.43 0.33 0.6 2.71 3.04

DH75 7.4 0.43 0.86 2.71 3.14

DH101 7.53 0.51 1.06 2.76 3.27

IH147 7.96 0.36 0.75 2.86 3.22

114 Chapter 3 Heat Capacity, Entropy, and Nanothermodynamics

Ecs(N ) values. However, due to the high pressure existing in the cluster, Eci(N ) would also increase. According to Table 3.6, the increasing extent of Eci(N ) is large than Ecs(N ), which induces the decreasing of corresponding γsv(N ). Since both cohesive values of surface Ecs(N ) and interior Eci(N ) would change for clusters, γsv(N ) is different from the corresponding bulk value γsv(∞).

3.6 Size Effect on Bandgap of II-VI Semiconductor

In document Qing Jiang Zi (pagina 127-133)