Thermodynamic Properties of Supported and Embedded Metallic Nanocrystals: Gold on/in SiO2
© to the authors 2008
Received: 17 July 2008
Accepted: 17 September 2008
Published: 9 October 2008
We report on the calculations of the cohesive energy, melting temperature and vacancy formation energy for Au nanocrystals with different size supported on and embedded in SiO2. The calculations are performed crossing our previous data on the surface free energy of the supported and embedded nanocrystals with the theoretical surface-area-difference model developed by W. H. Qi for the description of the size-dependent thermodynamics properties of low-dimensional solid-state systems. Such calculations are employed as a function of the nanocrystals size and surface energy. For nanocrystals supported on SiO2, as results of the calculations, we obtain, for a fixed nanocrystal size, an almost constant cohesive energy, melting temperature and vacancy formation energy as a function of their surface energy; instead, for those embedded in SiO2, they decreases when the nanocrystal surface free energy increases. Furthermore, the cohesive energy, melting temperature and vacancy formation energy increase when the nanocrystal size increases: for the nanocrystals on SiO2, they tend to the values of the bulk Au; for the nanocrystals in SiO2 in correspondence to sufficiently small values of their surface energy, they are greater than the bulk values. In the case of the melting temperature, this phenomenon corresponds to the experimentally well-known superheating process.
KeywordsNanocrystal Surface energy Gold SiO2 Cohesive energy Melting temperature Vacancy formation energy
The physical and chemical properties of low-dimensional solid-state systems draw considerable attention, in the previous years, because of their technological importance [1, 2]. In particular, the properties of nanocrystals (NCs) differ from that of the corresponding bulk materials, mainly due to the additional energetic term of γ 0 A, i.e. the product of the surface excess free energy γ 0 and the surface area A. This term becomes significant to change the physical and chemical properties of the NCs (with respect to the bulk material) due to the large surface/volume ratio of such systems. So, the properties of NCs bridge those of bulk materials and atomic scale systems . Also, the thermodynamic properties of NCs are different from that of the corresponding bulk materials and the study of such properties acquired a fundamental relevance in the last decades [4–34] because of their applications in the field of microelectronics, solar energy utilization and nonlinear optics. For example, nowhere is the interest in the thermodynamics of materials at small dimensions than in the microelectronics industry, where transistors and metal interconnects will have tolerances of only several nanometres . One particular phenomenon of interest is, for example, the size-dependent melting point of NCs with respect to the corresponding bulk materials [4–11]: this phenomenon received considerable attention since Takagi in 1954 experimentally demonstrated that ultrafine metallic NCs melt below their corresponding melting temperature . It is now known that the melting temperature of all low-dimensional solid-state systems (NCs, nanowires and nanosheets), including metallic [5–8], organic [9, 10] and semiconductor  depends on their size. For free standing NCs, the melting temperature decreases as its size decreases, while for NCs supported on or embedded in a matrix, they can melt below or above the melting point of the corresponding bulk crystal depending on the interface structure between the NCs and the surrounding environment (the substrate or the matrix) [12–18]. If the interfaces are coherent or semi-coherent, an enhancement of the melting point is present: this phenomenon is called superheating [12–14, 16, 18]. Otherwise, there is a depression of the melting point [16–18]. Not only the melting temperature but also several thermodynamic parameters, such as cohesive energy (the energy needed to divide the crystal into isolated atoms) and vacancy formation energy (the energy to form a vacancy in the NC in thermodynamic equilibrium) of NCs have been the subject of several experimental  and theoretical [18, 20–28] studies. Several methods have been developed and used to measure the thermodynamic properties of nanosystems (nanocalorimetry , scanning probe microscopy [29, 30], transmission electron microscopy , X-ray diffraction ) and different models were developed to describe the size-dependent thermodynamic properties of such systems (BOLS model , latent heat model , liquid drop model , surface-area-difference (SAD) model [22, 23], bond energy ). So, the importance of the combination of experimental and theoretical works towards a complete understanding of the thermodynamic properties of free-standing, supported and embedded nano-systems is clear.
In a previous experimental work , we fabricated supported/embedded Au NCs on/in SiO2. Using high-resolution transmission electron microscopy (HR-TEM) and by the inverse Wulff construction, we were able to obtain the angular-dependent surface energy γ 0 (θ) of the supported and embedded NCs as a function of their size. The main result of that work was that growing NCs surrounded by an ‘isotropic’ environment (Au in SiO2) exhibit an angular and radial symmetrical surface free energy function γ 0 (θ) making the surface stress tensor f ij of the NC rotationally and translationally invariant: this situation determines a symmetrical equilibrium shape of the NCs. Growing NCs in a ‘non-isotropic’ environment (Au on SiO2) exhibit a surface energy γ 0(θ), that lost its angular and radial symmetry for sufficiently large sizes, determining the loss of the rotational and translational invariance of the f ij and, as a consequence, a loss of symmetry in the equilibrium shape of the NC and the formation of internal defects.
In the present paper, we cross our experimental data concerning the surface free energy of the Au NCs supported on SiO2 and embedded in SiO2 with the SAD theoretical model of Qi  for the description of the size-dependent thermodynamic properties of metallic NCs, to simulate (and so to predict) the cohesive energy, the melting temperature and the vacancy formation energy for the Au NCs supported on and embedded in SiO2. In particular, we compare such quantities for Au NCs with the same size but supported or embedded on/in the SiO2 layer finding significant differences derived from the different effects of the surrounding environment. We believe that our theoretical results can improve the understanding of the thermodynamic properties of NCs and can guide future experimental investigations.
Experiment and results
Cz- < 100 > silicon substrates (with resistivity ) were etched in a 10% aqueous HF solution to remove the native oxide, and subsequently annealed at 1,223 K for 15 min in O2 in order to grow an uniform, 10 nm thick, amorphous SiO2 layer. A 2-nm thick Au film was deposited (at room temperature) by sputtering on the SiO2 layer using an Emitech K550x Sputter coater apparatus (Ar plasma, 10−4 Pa pressure). In some samples, the Au layer was covered by a 3-nm thick SiO2 layer deposited (at room temperature) by sputtering using an AJA RF Magnetron sputtering apparatus (Ar plasma, 1.3 × 10−8 Pa pressure). Then, the Au/SiO2 and SiO2/Au/SiO2 samples were contemporary annealed in Ar ambient in the 873–1073 K temperature range and in the 5–60 min time range to obtain the growth of NCs [33, 34].
The samples were analyzed by HR-TEM (after mechanical polishing and final Ar ion milling) using a Jeol 2010F energy-filtered transmission microscope (EF-TEM), operating at 200 kV and equipped with a Gatan image filtering apparatus. Before the cross-sectional TEM analyses of the samples with Au on SiO2, a thin SiO2cap-layer (~3 nm) was deposited (by sputtering with the AJA RF Magnetron apparatus) on the samples in order to protect the NCs during the samples preparation.
From the HR-TEM images of the Au NCs embedded in SiO2, with diameter in the range 2 nm < D < 7 nm it is clear that they conserve an equilibrium shape symmetry (single icosahedral crystal) increasing the size [33, 34].
Differently, the Au NCs growing on SiO2, belong to three different groups on the basis of their equilibrium shape and internal atomic structure, as a function of their diameterD(in the range 2 nm < D < 7 nm for our case):
group 1: formed by NCs with a diameterD < 3 nm. They have a structure such as the Au NCs in SiO2, i.e. single icosahedral crystal;
group 2: formed by NCs with a diameter 3 nm < D < 4 nm. They have an icosahedral structure characterized by twins of the (111) atomic planes;
group 3: formed by the NCs with a radius 4 nm < D < 14 nm. They have a complicated decahedral multi-twinned (and lamellar) structure.
Theoretical model, simulations and discussion
Qi et al.  developed a SAD model to describe the thermodynamic properties of metallic NCs and several experimental confirmations of the model were presented . This model is based on the knowledge of the surface (or interface) free energy of the NC to simulate size-dependent thermodynamic properties of the NC such as cohesive energy, melting temperature and vacancy energy formation.
The cohesive energy is an important physical quantity that accounts for the bond strength of a solid, which equals the energy needed to divide the solid into isolated atoms by breaking all the bonds. It is also a fundamental quantity in determining other important thermodynamic properties such as the melting temperature of the solid and the vacancy formation energy . The cohesive energy of a bulk crystal is constant at a given temperature , because of the small surface to volume ratio. But this ratio becomes significant in low-dimensional systems such as NCs. The experimental size-dependence of the cohesive energy for low-dimensional systems for Mo and W NCs was reported for the first time in 2002 , while the size-dependence of the melting temperature has been known already for a long time [4, 5]. Also, the vacancy formation energy is size-dependent in low-dimensional systems . Vacancies are very important defects in materials, and have a remarkable effect on the physics of materials, such as electrical resistance and heat capacity. Therefore, the importance of knowing its dependence on the NCs size is clear.
The SAD model developed by Qi et al.  describes the size-dependent cohesive energy, melting temperature and vacancy energy formation for low-dimensional systems in an excellent way. Therefore, our aim is to implement such a model with our experimental data on the surface free energy of supported and embedded Au NCs on/in SiO2 to simulate (i.e. to predict) their cohesive energy, melting temperature and vacancy formation energy as a function of the surface free energy and size. To this aim, we recall briefly the results of the model, whose extensive description can be found in .
In these formulas, we have and . γ0 is the surface free energy of the crystal per unit area (J/m2) at 0 K. γ i is the interface energy (per unit area) between the crystal and the surrounding matrix [18, 23]. γ M is the surface energy (per unit area) of the surrounding matrix at 0 K. is a parameter that describes the coherence between the crystal and the matrix . q = 1 describes the case of a completely coherent interface, q = 0.5 a semi-coherent interface, q = 0 the non-coherent interface (is the same as that of the crystal with the free surface). The parameter α is a shape factor: it accounts for the shape difference between spherical and non-spherical NCs [18, 23, 26, 27]. It is defined as the ratio of surface area between non-spherical and spherical NCs in an identical volume. For spherical NCs, a = 1. is the interplanar distance of (hkl).
Equations 13 with the appropriate values of describe very well the size-dependent cohesive energy, melting temperature and vacancy formation energy of NCs and they are used to predict such thermodynamic quantities for several low-dimensional solid-state systems .
We use Eqs. 1–3in this work in connection with the experimental data on the surface free energy of Au NCs on and in SiO2 (Fig. 1) to simulate E cNC, T mNC, E vNC as a function of γ 0 (θ i ) and D for the supported and embedded NCs.
Figure 2 reports the results for the cohesive energy simulations: black, red and green points indicate the calculations for the cohesive energy, as a function of the surface free energy γ 0 (θ), for Au NCs supported on SiO2 with diameter ofD = 2.5 nm,D = 3.5 nm,D = 7 nm, respectively; blue, cyan and magenta points indicate the calculated values for Au NCs embedded in SiO2with diameter ofD = 2.5, 3.5, 7 nm, respectively.
For the Au NCs on SiO2, the cohesive energy, melting temperature and vacancy formation energy are almost constant with γ 0 (θ);
The constant values of the cohesive energy, melting temperature and vacancy formation energy for the Au NCs on SiO2 increase when the NCs size increases, tending to the values of the bulk Au;
For the Au NCs in SiO2, the cohesive energy, melting temperature and vacancy formation energy decrease when γ 0 (θ) increases;
The constant values of the cohesive energy, melting temperature and vacancy formation energy for the Au NCs in SiO2 increase when the NCs size increases;
For the smaller NCs in SiO2, in correspondence of opportune small values of γ 0 (θ), the values of the cohesive energy, melting temperature and vacancy formation energy are greater than the bulk values.
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