Size-induced undercooling and overheating in phase transitions in bare and embedded clusters

Chang Q. Sun, Yong Shi, C. M. Li, S. Li, and T. C. Au Yeung

Phys. Rev. B 73, 075408 – Published 3 February 2006

Abstract

The correlation between the thermal stability and electroaffinity of a nanosolid has been explored from the perspective of surface and interface bond-order deficiency. It turns out that the coherency of an atom at the grain boundary and the portion of atoms in the skin of a nanosolid dominate the size dependence of critical temperatures $(T_{C})$ for phase transitions. The trapping potential well depression at the surface and interface not only shifts the valence density of state positively but also enlarges the electroaffinity that determines the strength of the bond. In particular, bond-nature alteration at the junction interface, or bond-nature evolution with the reduction of atomic coordination of III- or IV-A atoms, dominates the irregular $T_{C}$ change with the sizes of the embedded or the III- or IV-A bare nanosolids. Atoms in “superficial” or “interfacial” skins play the core role in dictating the size effect on the thermal stability and electroaffinity of a nanosolid whereas atoms in the core interior remain as they are in the bulk.

Received 20 September 2005

DOI:https://doi.org/10.1103/PhysRevB.73.075408

Authors & Affiliations

Chang Q. Sun^1,*, Yong Shi², C. M. Li³, S. Li⁴, and T. C. Au Yeung¹

¹School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798
²East Campus, Tangshan University, Hebei Province, China 063020
³School of Chemical and Biomedical Engineering, Nanyang Technological University, Singapore 639798
⁴School of Materials Science and Engineering, University of New South Wales, Sydney NSW2052, Australia

^*FAX: 65 6792 0415. Electronic address: ecqsun@ntu.edu.sg

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Vol. 73, Iss. 7 — 15 February 2006

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Figure 1
Correlation between electroaffinity (separation between vacuum level and the conduction band edge) enlargement $Δ ε (K_{j})$ and conduction band shift $Δ E_{C} (K_{j})$ : $Δ ε (K_{j}) = Δ E_{C} (K_{j}) - Δ E_{G} (K_{j}) ∕ 2 - Δ E_{C} (\infty) = [Δ E_{C} (\infty) - E_{G} (\infty) ∕ 2] Δ_{H}$ , with $Δ_{H}$ being the perturbation to the Hamiltonian due to BOLS correlation (Ref. 31). $Δ E_{G} (K_{j})$ is the size-induced band gap expansion. The energy level drops with the increase of number atoms to the maximum (at $z = 2$ for a monatomic chain or a unit cell) and then rises in a $K_{j}^{- 1}$ way to the bulk value, $Δ E_{C} (\infty)$ . The shaded area corresponds to the electroaffinity enlargement.Reuse & Permissions
Figure 2
(Color online) Comparison of the measured size dependence of atomic cohesive energy $E_{B}$ for Mo and W nanosolids (Ref. 53) with three different models. Numerical agreement is nearly identical for the compared models whereas the physical indications are completely different.Reuse & Permissions
Figure 3
(Color online) Agreement between predictions (lines) and experimental observations (scattered symbols) of the size and shape dependence of the $T_{m}$ suppression of (a) Sn and Al on ${Si}_{3} N_{4}$ substrate (Ref. 56), (b) In (Ref. 57) and Pb (Ref. 58), (c) Bi (Ref. 57) and CdS (Ref. 59), (d) Ne (Ref. 60), Kr (Ref. 61), O, and methylchloride ( $m$ -CL, only one data point) (Ref. 62). Al nanosolids grown on SiN substrate are more plate like ( $τ = 1$ , $m = 1$ ) throughout the sizes of measurement [panel (a)] but Sn-01 on SiN [panel (a)] and In-01 and Pb-01 [panel (b)] are more spherical like $(τ = 3)$ at particle size smaller than $10 nm$ . Indium particles [panel (b)] encapsulated in pore silica matrix are slight overheating compared with Vycor glass matrix. The $T_{m}$ of Kr, Ne, and O solids follow the curve of $m = 4.88$ better despite the accuracy of measurement.Reuse & Permissions
Figure 4
(Color online) Agreement between BOLS predictions (curves) and measurements (scattered data) of $T_{C}$ suppression (a) for ferromagnetic Ni films (Ref. 63), and ${Fe}_{2} O_{3}$ grains (Ref. 64), and ferroelectric $Si B_{2} Ta O_{3}$ nanostructures (Ref. 65). The size offset of $K_{0}$ represents the contribution from long-order dipole interaction. (b) Liquid-vapor transition $(T_{vap})$ of Ag and PbS nanosolids (Ref. 66). Agreement gained so far (Figs. 2, 3) favors the universal trend of the relative change disregarding the actual processes of thermally stimulated phenomena despite the uncertainty being subject to accuracy of measurement.Reuse & Permissions
Figure 5
(Color online) Matching predictions to the measurements of (a) overheating of In (Ref. 67) and Pb (Ref. 69) embedded in Al matrix (Ref. 68), leads to the $α = z_{i b} c_{i}^{- m}$ values as indicated, which describes the interfacial bond strength. (b) Agreement between BOLS predictions with measurements (Ref. 82) of $T_{S} = 0.3 T_{m}$ dependence of critical sizes for W $[T_{m} (\infty) = 3695 K]$ , Ni $(1728 K)$ , and Ag $(1235 K)$ nanocrystal growth.Reuse & Permissions

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