On the brittleness of elementary semiconductors
At a Glance
Section titled “At a Glance”| Metadata | Details |
|---|---|
| Publication Date | 2023-01-01 |
| Journal | Физика твердого тела |
| Authors | М. Н. Магомедов |
| Institutions | Joint Institute for High Temperatures |
| Citations | 1 |
| Analysis | Full AI Review Included |
On the Brittleness of Elementary Semiconductors: An Engineering Analysis
Section titled “On the Brittleness of Elementary Semiconductors: An Engineering Analysis”Executive Summary
Section titled “Executive Summary”This paper presents a novel analytical model explaining the inherent brittleness and the Brittle-Ductile Transition (BDT) in elemental covalent crystals (Diamond, Silicon, Germanium) without relying on computer simulations.
- Core Mechanism: Brittleness is caused by the “duplicity” of the paired interatomic potential. The potential depth for elastic (reversible) deformation ($D_b$) is distinct from the depth for plastic (irreversible) deformation ($D_s$).
- Energy Advantage for Fracture: The specific surface energy required for plastic fracture ($\sigma_s$) is calculated to be more than two times less than the energy required for elastic stretching ($\sigma_e$). This makes brittle fracture energetically favorable at low strain.
- Brittle-Ductile Transition (BDT) Trigger: The transition from brittle to ductile behavior ($T_{BDT}$) is fundamentally linked to the metallization of the paired covalent bonds on the crystal surface.
- Analytical BDT Calculation: A new analytical method was developed to calculate $T_{BDT}$ based on thermodynamic equilibrium and surface energy ratios.
- Temperature Limit: The theoretical upper limit for the BDT temperature ratio under static load conditions is established: $T_{BDT}/T_m < 0.45$, where $T_m$ is the melting temperature.
- Model Scope: The approach is applicable not only to single crystals but potentially to metastable states like glass or nanostructured materials.
Technical Specifications
Section titled “Technical Specifications”Data extracted from the analytical model and experimental validation for key elemental semiconductors.
| Parameter | Value | Unit | Context |
|---|---|---|---|
| C-dia Elastic Potential Depth ($D_b$) | 8.43 | eV | Used for reversible elastic deformation |
| C-dia Plastic Potential Depth ($D_s$) | 3.68 | eV | Used for irreversible plastic fracture |
| Si Elastic Surface Energy ($\sigma_e$) | 4001.10 | 10-3 J/m2 | Calculated at 0 K using $D_b$ |
| Si Plastic Surface Energy ($\sigma_s$) | 1673.42 | 10-3 J/m2 | Calculated at 0 K using $D_s$ |
| Si Energy Ratio ($\sigma_e / \sigma_s$) | 2.391 | Dimensionless | Elastic energy is 2.39 times plastic energy |
| Ge Energy Ratio ($\sigma_e / \sigma_s$) | 2.078 | Dimensionless | Elastic energy is 2.08 times plastic energy |
| C-dia Calculated $T_{BDT}/T_m$ | 0.437 | Dimensionless | Calculated using liquid phase $\sigma_s$ estimate |
| Si Calculated $T_{BDT}/T_m$ | 0.442 | Dimensionless | Calculated using liquid phase $\sigma_s$ estimate |
| Ge Calculated $T_{BDT}/T_m$ | 0.426 | Dimensionless | Calculated using liquid phase $\sigma_s$ estimate |
| Theoretical $T_{BDT}/T_m$ Upper Limit | < 0.45 | Dimensionless | Under static load (infinitely low deformation rate) |
| C-dia Experimental $T_{BDT}$ Range | 1470 - 1615 | K | Measured BDT range |
| Si Experimental $T_{BDT}$ Range | 820 - 1225 | K | Measured BDT range |
Key Methodologies
Section titled “Key Methodologies”The study utilized a purely analytical approach based on fundamental solid-state physics principles, avoiding computational simulations.
- Interatomic Potential Definition: The Mie-Lennard-Jones potential (Equation 1) was adopted to model the pair interatomic interaction. Parameters ($r_0$, $b$, $a$) were derived from macroscopic properties (molar volume, Grüneisen parameter, modulus of elasticity).
- Potential Duplicity Implementation: Two distinct potential depths were defined based on the type of deformation:
- $D_b$: Total bond energy, used for reversible (elastic) deformation parameters (e.g., sound speed, Debye temperature).
- $D_s$: Weak single bond energy ($L_{00}/k_n$), used for irreversible (plastic) deformation parameters (e.g., specific surface energy, sublimation energy).
- Fracture Initiation Condition: The prerequisite for brittle fracture was established as an energy inequality: $\Delta E = (\sigma_e - \sigma_s) \delta S \ge 0$. This confirms that forming a surface via irreversible breaking ($\sigma_s$) is energetically cheaper than via elastic stretching ($\sigma_e$).
- Surface Energy Calculation: The specific face surface energy ($\sigma(100)$) was calculated using an expression (Equation 7) derived from the interatomic potential, the “only nearest neighbors interaction” approximation, and the Einstein model for crystal vibration.
- BDT Temperature Derivation: An analytical expression for $T_{BDT}/T_m$ (Equation 15) was derived by setting the condition for BDT: $\sigma_e(T_{BDT}) = 2\sigma_s(T_{BDT})$. This derivation incorporated the temperature dependence of surface energy and the condition that $\sigma_e(T_m) = \sigma_s(T_m)$ (full covalent bonding at melting).
Commercial Applications
Section titled “Commercial Applications”This research provides critical theoretical insight for optimizing manufacturing processes and predicting material reliability in industries relying on elemental semiconductors and ultra-hard materials.
- Semiconductor Wafer Processing: The model explains why Si and Ge wafers are prone to brittle fracture during cutting, grinding, and polishing below $T_{BDT}$. This knowledge is essential for setting optimal processing temperatures and strain rates to ensure ductile behavior and minimize defects.
- High-Power Electronics and Thermal Management: Understanding the BDT mechanism (metallization) helps predict the failure points of Si and SiC devices under extreme thermal and mechanical stress, crucial for reliability in high-voltage and high-frequency applications.
- Diamond Tooling and Coatings: For C-dia (diamond), the high $T_{BDT}$ (1470-1615 K) confirms its extreme brittleness. This research aids in designing synthesis and post-processing techniques to enhance toughness without compromising hardness.
- Nanomaterials and Glass Science: The theoretical framework is suggested to apply to metastable materials (like glass or nanostructured crystals), offering a pathway to predict and control the fracture behavior of new amorphous or polycrystalline structures.
- Reliability Engineering: The established upper limit for $T_{BDT}/T_m$ provides a fundamental constraint for material selection and operational temperature limits in structural components where catastrophic brittle failure must be avoided.
View Original Abstract
It is shown that the brittleness of a single-component covalent crystal (diamond, Si, Ge) is due to the &quot;duplicity&quot; of the paired potential of interatomic interaction for elastic (reversible) and for plastic (irreversible) deformation. This leads to the fact that the specific surface energy during plastic deformation of a covalent crystal is more than two times less than the specific surface energy during elastic deformation. Therefore, with a small deformation of a covalent crystal, it is energetically more advantageous to create a surface by irreversible breaking than by reversible elastic stretching. It is indicated that the brittle-ductile transition in a single-component covalent crystal is accompanied by metallization of covalent bonds on the surface. It is shown that the brittle-ductile transition temperature (T BDT ) for single-component covalent crystals under static load has an upper limit: T BDT /T m &lt;0.45, where T m --- is the melting temperature. Keywords: interatomic covalent bond, brittleness, ductility, elementary semiconductors, brittle-ductile transition.