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than 0.01 eV/ Å. To sample the Brillouin zone, we have utilized 242424 Monkhorst-Pack scheme46 for the equilibrium state, deformed structure, and calculating elastic tensor (Hessian matrix which is the second derivatives of the energy with respect to the atomic positions). Here, for convenient notation, we have assigned x-, y-, and z-directions to the [100]-, [010]-, [001]-directions.
3.3 Equilibrium state of covalent materials
In this Section, we have focused on the mechanical properties of covalent materials (C, Si, and Ge) using Hill’s criterion to determine the ideal strength of materials . Before applying the uniaxial stress, the equilibrium state of covalent materials (diamond structure including 8 atoms) is calculated and summarized in Figure 3.1 to find lattice constant around 3.536, 5.403, and 5.647 Å for C, Si, and Ge bulk, respectively. And, its corresponding cohesive energies (eV/atom) and elastic constants (GPa) at equilibrium state are also shown in Table 3.1. Here, the cohesive energy is obtained from the relation of Ebulk −Eatom, where Ebulk is the total energy per atom of bulk material, Eatom is the total energy of isolated single atom. Our results on basis properties of covalent materials at equilibrium state are in good agreement with previous studies65–68. In case of carbon, known as the hardest material, the overall elastic constants (C11, C12, and C44) is much higher than other covalent materials. In Figure 3.2, we have confirmed the directional charge distribution (sp3 hybrid orbitals) of C, Si, and Ge bulk at equilibrium state, which is one of unique characteristics of covalent materials. The elastic stabilities of diamond materials are also confirmed and summarized in Table 3.2. Based on the optimized lattice parameter, we have computed the elastic stability condition terms for cubic symmetry with C11 +2C12,
11 12
C −C , and C44, so-called Born stability criterion which are related to the bulk, shear, and tetragonal shear moduli, respectively. As can be seen in Table 3.2, the overall elastic stability condition terms show positive values, indicating that the optimized diamond configurations are stable in terms of Hill’s criterion.
3.4 Mechanical response under uniaxial loading and elastic stability
In order to investigate the mechanical properties of covalent materials, we have considered the uniaxial loading condition along x-direction as shown in Figure 3.3a. Under uniaxial tensile stress, the other transverse directions (y- and z-directions) are fully relaxed until to be almost zero stress (in GPa).
Because of tensile stress along the [100]-direction, there is the change in transverse lattices to be relatively stable configuration (In this case, the compressive strain is generated along transverse
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directions). Interestingly, after 0.13 of tensile strain to C diamond, the transverse lattices are elongated so-called negative Poisson’s ratio as shown in Figure 3.3b. At this strain, the elastic constant of C13 becomes negative value. While, for Si and Ge, the lattices along the transverse directions are gradually decreased as applying tensile stress along loading direction. Also, we have seen that the cohesive energy of materials is gradually increased as shown in Figure 3.4.
Based on those configurations, the elastic stability condition terms of tetragonally deformed covalent materials are estimated as shown in Figure 3.5. As increasing tensile strain, all elastic stability condition terms are gradually increased. And then, we have obtained the critical strains for instability of C, Si, and Ge bulk around 0.34, 0.22, and 0.23 of tensile strain, respectively. Compared bulk noble metals (0.11, 0.10, and 0.08 of tensile strain for Cu, Ag, and Au, respectively), the instabilities of covalent materials are late initiated in terms of elastic theory. In case of C diamond, the elastic stability condition terms of T5 and T4 touch zero at same strain. For Si bulk, the elastic stability condition term of T3 (at 0.22 of tensile strain) become negative value, and then T2 (at 0.23 of tensile strain) could be predicted as another elastic instability. The elastic stability condition terms of T2 and T3 touches zero at 0.23 and 0.25 of tensile strain for Ge bulk, respectively. Unlike bulk noble metals, the elastic stability condition terms (at least two) of covalent materials have been investigated at the similar tensile strain.
The corresponding deformation mode would be discussed in Section 3.5.
3.5 Phase transformation at unstable state
Figure 3.6 shows that the phase transformation of C bulk happens from the diamond (sp3 hybrid orbital at A-point of Figure 3.6) to graphite configuration (sp2 hybrid orbital at B-point) during relaxation of material. This phenomenon is previously reported by DFT calculation62 and nano-indentation of experiment64. Here, for this, we have slightly applied the perturbation to 18%-tensioned C bulk, and then performed the optimization (There is constraint along x-direction.) to obtain another stable configuration. This result is not matched with the critical strain obtained from elastic criterion (in Figure 3.5a), indicating that for C diamond under uniaxial loading the elastic criterion would not be correct to predict the unstable state. We might believe that the phonon instability is observed in short-wavelength around 0.18 of tensile strain (Here, we would not utilize the phonon criterion). In addition, there is complicated loading to induce phase-transformation from diamond to graphite structure, which could be not predicted by deformation mode (introduced in Eq. (1.55)-(1.58)) of elastic stability. We could say that there is difficulty to use elastic criterion for allotrope materials. However, for Si and Ge bulk, the phase-transformation is observed like bifurcation of noble metals (See Figure 2.10) as expected by
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the elastic stability condition term of T3. Interestingly, the amount of change (increase and decrease in transverse lattices) is not symmetry as observed in noble metals. We have seen that at unstable state expected by elastic stability, the bifurcation of Si and Ge bulk is asymmetrically shown in Figure 3.7a and 3.8a, respectively. These results indicate that Hill’s criterion could be utilized also for predicting the instability of Si and Ge bulk. Unlike noble metals, the amount of stress change for Si and Ge bulk along x-direction is slightly decreased less than 500MPa after phase-transformation (bifurcation) as shown in Figure 3.7b and 3.8b, respectively. In case of Ge bulk, we have found another instability around 0.26 of tensile strain based on the elastic stability condition term of T5 as shown in Figure 3.8c.
One of angle (between x- and y-directions (or z-direction) is gradually changed, which is energetically stable. We have shown that elastic criterion could well detect unstable configurations and corresponding deformation mode under uniaxial tensile loading.
3.6 Effect of symmetrical transverse loading
Here, we have applied additional transverse loading to Si and Ge bulk to confirm the trend of the elastic stability condition terms and corresponding ideal strength of material. To investigate the effect of transverse loading, the critical strain along the x-direction is initially applied to structure, which is obtained from Hill’s criterion shown in Figure 3.5 for Si (0.22 of tensile strain) and Ge (0.23 of tensile strain) bulk. The tensile strain around 0.22 along x-direction is applied, and then additionally symmetrical transverse stress is applied to Si bulk. The trend of elastic stability condition terms with different transverse loading is shown in Figure 3.9a. As applying additional compressive strain to transverse directions based on the equilibrium state of Si bulk under 0.22 of tensile strain (denoted by orange dashed line in Figure 3.9a), almost elastic stability terms become negative value. While, under additional transverse tensile strain to Si bulk, the elastic stability condition term of T3 is enhanced to be positive value, indicating that the 22%-tensioned Si bulk becomes elastically stable. Also, the stress along the [100]-direction is estimated with different transverse loading as shown in Figure 3.9b.
Because of initiation of instability nearby theoretical strength, there is no great effect in increasing ideal strength of Si bulk even applying additional tensile transverse strain unlike noble metals (See Figure 2.17d). For 23%-tensioned Ge bulk, the elastic stability condition terms are mostly enhanced with applying additional tensile transverse stress as shown in Figure 3.10a. Also, the ideal strength of Ge bulk is not significantly increased as applying transverse stress as shown in Figure 3.11b. Here, we have inferred that the late instability in terms of strain is related to the characteristic bonding and/or diamond configuration, resulting in slight effect of transverse loading to ideal strength of Si and Ge bulk. These
34 assumptions will be discussed as our next study.
3.7 Summary
In this Section, we have focused on the mechanical property of covalent materials (C, Si, and Ge) with support of elastic stability criterion. In case of diamond C under uniaxial loading, there is phase- transformation from diamond to graphite structure at 0.18 of tensile strain, which could not be predicted by Hill’s criterion (0.33 of tensile strain). However, for Si and Ge bulk, the homogeneous deformation could be well detected by the elastic stability condition term of T3. At unstable state, the Si (0.22 of tensile strain) and Ge (0.23 of tensile strain) bulk have been transformed from tetragonal to orthorhombic system. In addition, for Ge bulk, another instability is also found by the elastic stability condition term of T5, resulting in the change of angle between x- and y-direction (or z-direction). Also, we have seen that the elastic stability could be enhanced by transverse loading for Si and Ge bulk.
However, unlike noble metals (Cu, Ag, and Au bulk), the ideal strength is not significantly increased, which could be thought of the late initiation of instability.
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