Thermodynamic Theory of Phase Separation in Nonstoichiometric Si Oxide Films Induced by High-Temperature Anneals
Abstract
:1. Introduction
2. Theory and Results
2.1. Gibbs Free Energy of Si/Si Oxide Systems
- The Gibbs free energy of the a-Si phase is expressed in terms of the Gibbs free energy of c-Si plus the excess energy equal to per one Si atom [51]. Here, hE = 13,400 J/mole is the molar crystallization enthalpy [52], sE = 3.97 J/mole × K is the molar excess entropy of amorphous-to-crystalline transition [53], and NA = 6.022 × 1023 mole−1 is the Avogadro constant, respectively.
- The nonstoichiometric Si oxide phase is considered in the binary solution approximation [54], in which the SiOx solution is formed by mixing elemental Si and oxygen atoms obtained by the decomposition of O2 molecules.
- The structure of Si oxide is made up of interconnected Si–OySi4–y tetrahedral units (0 ≤ y ≤ 4) with different oxidation degrees y of a central Si atom. In addition to the Si–Si and Si–O bond energies, each tetrahedral unit is characterized by the penalty energy Δy as the measure of the energy nonequivalence of the units with different values of y (Δ0 = Δ4 = 0 eV, Δ1 = 0.5 eV, Δ2 = 0.51 eV, and Δ3 = 0.22 eV) [55]. The probability of finding a tetrahedral unit with either value of y obeys the random bonding model proposed by Phillip [56].
- The entropy of mixing is considered to be the configuration entropy of the Si oxide phase, associated with the number of arrangements of oxygen atoms between the pairs of Si atoms.
2.2. Effect of Nano-Si/Si Oxide Matrix Interfaces on Phase Equilibria in Si/Si Oxide Systems
2.3. Influence of Internal Stress on Phase Separation in SiOx Films
- the superlinear dependence on the difference between the current and the initial Si oxide stoichiometry indexes, to be able to reproduce the increase in the value of xeq with the increase in x0;
- the descending dependence on the annealing temperature, to enable the minimum Gibbs free energy of a Si/Si oxide system shift toward smaller x, upon raising the temperature.
2.4. Crystallization Model of Amorphous Si Nanoinclusions Embedded in the Si Oxide Matrix
- The crystallization of an amorphous Si nanoinclusion is incomplete, and an amorphous shell between the crystalline core and the Si oxide matrix always remains. This result is illustrated by the exemplary dependences of Δgcryst on ξ for different initial radii R at 900 °C presented in Figure 8a, in which ξmin corresponds to the equilibrium states.
- A minimum radius of amorphous Si nanoinclusions exists, below which crystallization becomes energetically unfavorable. The value of this radius increases with the increase in annealing temperature. The dependence of the Gibbs free energy change on ξ for a Si nanoinclusion, with the radius below the crystallization threshold, is illustrated by the curve (1) in Figure 8.
- The crystallized Si fraction (value of ξmin) increases, and the nucleation barrier for crystallization decreases, with the increase in a-Si nanoinclusion size, due to the weakening of the influence of the Si oxide matrix on the crystallization process. The saturation of the nucleation barrier at R ≈ 2 nm points to an almost complete loss of this influence. This result is illustrated by the dependences presented in Figure 9a,b for the crystallization temperature of 900 °C (see also Figure 8a,b).
3. Discussion
4. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
References
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Sarikov, A. Thermodynamic Theory of Phase Separation in Nonstoichiometric Si Oxide Films Induced by High-Temperature Anneals. Nanomanufacturing 2023, 3, 293-314. https://doi.org/10.3390/nanomanufacturing3030019
Sarikov A. Thermodynamic Theory of Phase Separation in Nonstoichiometric Si Oxide Films Induced by High-Temperature Anneals. Nanomanufacturing. 2023; 3(3):293-314. https://doi.org/10.3390/nanomanufacturing3030019
Chicago/Turabian StyleSarikov, Andrey. 2023. "Thermodynamic Theory of Phase Separation in Nonstoichiometric Si Oxide Films Induced by High-Temperature Anneals" Nanomanufacturing 3, no. 3: 293-314. https://doi.org/10.3390/nanomanufacturing3030019