Configurational entropy significantly influences point defect thermodynamics and diffusion in crystalline silicon

It has long been suggested that the familiar intrinsic point defects (vacancies and self-interstitials) encountered in crystals at low temperatures ($T$) transform into extended domains characterized by a missing or excess atom compared with the same-sized region in the perfect crystal so that such extended defects may be viewed as dropletlike regions of enhanced or diminished density. However, the implications of such a transformation, or whether it even occurs in crystalline Si, remain uncertain. To address this fundamental problem, we consider a comprehensive thermodynamic analysis of the thermodynamics of vacancy and self-interstitial formation over a broad $T$ range based on thermodynamic integration with a focus on entropic contributions. In cooled liquids, it is well known that the form of the intermolecular potential can greatly influence the configurational entropy $S_c$, and correspondingly, we analyze several empirical Si potentials to determine how the potential influences both the $T$ dependence of $S_c$ and the enthalpy and entropy of defect formation. We indeed find that the $S_c$ associated with point defects increases significantly upon heating, consistent with the existence of extended defects. Moreover, each type of defect species gives a significantly different contribution to $S_c$ at elevated $T$ and to a qualitive difference in the $T$ dependence of the entropy of defect formation in the extended defect regime. We discuss some potential consequences of these thermodynamic changes of defect formation on the $T$ dependence of diffusion in heated crystals.

Figure 1

Cartoon representation of the potential energy landscape (PEL) of a bulk crystal phase. Without surfaces, the crystal must nucleate its own defects, which eventually drive homogeneous melting (upper, blue-shaded band). Finite crystals bounded by surfaces can incorporate isolated point defects, which introduce their own contribution into the PEL (lower, orange-shaded band).

Figure 2

Formation thermodynamic properties of isolated self-interstitials (left) and vacancies (right) as a function of temperature computed with four different empirical potentials. (a) and (b) Total formation free energy, (c) and (d) average formation energy, and (e) and (f) total defect formation entropy.

Figure 3

Example relaxed configurations of self-interstitials (top) and vacancies (bottom) predicted by the environment-dependent interatomic potential (EDIP), along with their corresponding formation energies. The self-interstitial configuration at $\mathrm{\Delta }{E}^I=3.77$ eV corresponds to the $\langle 110\rangle$ dumbbell, the vacancy configuration at $\mathrm{\Delta }{E}^V=3.22$ eV is the vacancy ground state, and the vacancy configuration at $\mathrm{\Delta }{E}^V=3.77$ eV is the split vacancy. Color coding represents atomic energy: red > orange > yellow > green > cyan > blue.

Figure 4

Formation entropy components of isolated self-interstitials (left) and vacancies (right) as a function of temperature. (a) and (b) Configurational entropy and (c) and (d) total vibrational entropy.

Figure 5

Top: Anharmonic vibrational entropy of formation for (a) isolated self-interstitials and (b) vacancies as a function of temperature. Bottom: Schematic representation of collective basin occupancy in (c) low and (d) high configurational entropy settings. Blue shading in basins denotes the probability of occupancy.

Figure 6

Diffusivities and self-diffusion activation energies for self-interstitials (left) and vacancies (right) as a function of temperature. (a) and (b) Diffusion coefficients and (c) and (d) self-diffusion activation energies.