Phase stability, structural and mechanical properties of transition metal nitrides using first principles calculations

Abstract

The focus of this thesis is on understanding the phase stability as well as the structural and mechanical properties of transition metal nitrides using first principles calculations including detailed studies on (i) the stabilization of the cubic phase of WN, (ii) the phase formation of Mo 1-x N x as a function of x, (iii) the influence of the valence electron concentration on the mechanical stability of all cubic transition metal nitrides, and (iv) the energetics of point defects of twelve transition metal nitrides.

Influence of valence electron concentration on the mechanical properties: First principle calculations are employed to determine the mechanical properties of rock-salt structure binary and ternary transition metal nitrides, carbides, and carbonitrides from group IV B to XII B, predicting a unified indicator for mechanical properties: the valence electron concentration VEC. Pugh’s and Poisson’s ratios indicate an increasing ductility with increasing VEC, with a brittle-to-ductile transition at a critical VEC = 9. xiiThe calculated C 44 of binary and ternary carbonitrides and metallic alloy nitrides monotonously decreases from 164 ± 12 GPa at VEC = 8 to – 39 ± 46 GPa at VEC = 11, indicating a transition to mechanical instability at VEC = 10.65. Similarly, the isotropic elastic modulus decreases slightly from 420 GPa for VEC = 8 to 388 GPa, for VEC = 10 and finally to -98 GPa at a VEC of 11 and calculated hardness decreases from an average of 25 GPa for VEC = 8 to 12 GPa for VEC = 10 to 2 GPa for VEC = 11. Interpolation of the calculated C 44 of the ternary transition metal carbonitrides and nitrides of metallic alloys indicate a critical VEC of 10.65 ± 0.15 at which transition from mechanical stability to instability occurs. Phonon dispersion curves exhibit imaginary frequencies for VEC > 10, indicating a transition from dynamical stability-to- instability at a valence electron concentration, which is smaller than the critical VEC = 10.65 for the mechanical stability-instability transition.

Energetics of point defects in twelve transition metal nitrides: First principle calculations are employed to calculate the energetics of point defects – vacancies, antisites and interstitials, in group III B – VI B transition metal nitrides in rocksalt structure. The formation enthalpies of the point defects indicate that vacancies are the most preferred defects energetically among the point defects. The formation enthalpy of the anion vacancies decrease steadily from an average of 2.7 eV for group III B nitrides to -5.5 eV for group VI B nitrides and thermodynamically stable for group VI B nitrides. Cation vacancies in contrast decrease steeply from an average of 4.5 eV for group III B nitrides to -0.8 eV for group IV B nitrides and finally to -2.8 eV for group VI B nitrides, thus becoming thermodynamically stable for group IV B – VI B nitrides. Antisite defects of both kinds possess a high formation enthalpy indicating that these defects are unlikely to form. Cation interstitials similarly, exhibits a high formation enthalpy for group III B to V B nitrides decreasing from an average of 9.2 eV to 8.2 eV and decreases steeply to an average of -3.2 eV for group VI nitrides.

The anion interstitials, similarly, decreases from an average of 3.5 eV for group III B nitrides to an average of 1.6 eV for group V B nitrides before decreasing steeply to -7.7 eV for group VI B nitrides. The thermodynamic favorability for the formation of interstitials and vacancies in group VI B nitrides indicates the mechanical and thermodynamical instability of group VI B nitrides. In addition, the role of entropy of nitrogen gas is studied and the chemical potential of nitrogen is found to decrease with increasing temperature and pressure causing the anion vacancies, cation antisites and cation interstitials to become more favorable thermodynamically with increasing temperature while cation vacancies, anion antisites and anion interstitials become less favorable with increasing temperature.

The volume per atom V o of the thermodynamically stable Mo 1-x N x phases decreases from 13.17 to 9.56 Å 3 as x increases from 0.25 to 0.67, with plateaus at V o = 11.59 Å 3 for hexagonal and cubic phases and V o = 10.95 Å 3 for orthorhombic and monoclinic phases. The plateaus are attributed to changes in the average coordination numbers of molybdenum and nitrogen atoms, which increase from 2 to 6 and decrease from 6 to 4, respectively, indicating an increasing covalent bonding character with increasing x. The change in bonding character and the associated phase change from hexagonal to cubic/orthorhombic to monoclinic cause steep increases in the isotropic elastic modulus E = 387 – 487 GPa, the shear modulus G = 150 – 196 GPa, and the hardness H = 14 – 24 GPa in the relatively narrow composition range x = 0.4 – 0.5. This also causes a drop in Poisson’s ratio from 0.29 to 0.24 and an increase in Pugh’s ratio from 0.49 to 0.64, indicating a ductile-to-brittle transition between x = 0.44 – 0.5.

Phase stability of Mo 1-x N x system: First-principles density-functional calculations coupled with the USPEX evolutionary phase-search algorithm are employed to calculate the convex hull of the Mo-N binary system. Eight Mo 1-x N x compound phases are found to be thermodynamically stable: tetragonal β-Mo 3 N, hexagonal δ-Mo 3 N 2 , cubic γ- Mo 11 N 8 , orthorhombic ε-Mo 4 N 3 , cubic γ-Mo 14 N 11 , monoclinic σ-MoN and σ-Mo 2 N 3 and hexagonal δ-MoN 2 . The convex hull is a straight line for 0 ≤ x ≤ 0.44 such that bcc Mo and the five listed compound phases with x ≤ 0.44 are predicted to co-exist in thermodynamic equilibrium. Comparing the convex hulls of cubic and hexagonal Mo x N 1-x indicates that cubic structures are preferred for molybdenum rich (x < 0.3) compounds, hexagonal phases are favored for nitrogen rich (x > 0.5) compositions, while similar formation enthalpies for cubic and hexagonal phases at intermediate x = 0.3 – 0.5 imply that kinetic factors play a crucial role in the phase formation.

Stabilization of the cubic phase of WN: First-principles methods are employed to determine the structural, mechanical, and thermodynamic reasons for the experimentally reported cubic WN phase. The defect-free rocksalt phase is both mechanically and thermodynamically unstable, with a negative single crystal shear modulus C 44 = −86 GPa and a positive enthalpy of formation per formula unit H f = 0.623 eV with respect to molecular nitrogen and metallic W. In contrast, WN in the NbO phase is stable, with C 44 = 175 GPa and H f = −0.839 eV. A charge distribution analysis reveals that the application of shear strain along [100] in rocksalt WN results in an increased overlap of the t 2g orbitals which causes electron migration from the expanded to the shortened W-W [110] bond axes, yielding a negative shear modulus due to an energy reduction associated with new bonding states 8.1–8.7 eV below the Fermi level. A corresponding shear strain in WN in the NbO phase results in an energy increase and a positive shear modulus. The mechanical stability transition from the NaCl to the NbO phase is explored using supercell calculations of the NaCl structure containing C v = 0% − 25% cation and anion vacancies, while keeping the N-to-W ratio constant at unity. The structure is mechanically unstable for C v < 5%. At this critical vacancy concentration, the isotropic elastic modulus E of cubic WN is zero, but increases steeply to E = 445 GPa for C v = 10%, and then less steeply to E = 561 GPa for C v = 25%. Correspondingly, the hardness estimated using Tian’s model increases from 0 to 15 to 26 GPa as C v increases from 5% to 10% to 25%, indicating that a relatively small vacancy concentration stabilizes the cubic WN phase and that the large variations in reported mechanical properties of WN can be attributed to relatively small changes in C v .