Nitrogen vacancy, self-interstitial diffusion, and Frenkel-pair formation/dissociation in $B1$ TiN studied by ab initio and classical molecular dynamics with optimized potentials

We use ab initio and classical molecular dynamics (AIMD and CMD) based on the modified embedded-atom method (MEAM) potential to simulate diffusion of N vacancy and N self-interstitial point defects in $B1$ TiN. TiN MEAM parameters are optimized to obtain CMD nitrogen point-defect jump rates in agreement with AIMD predictions, as well as an excellent description of $\mathrm{Ti}{\mathrm{N}}_x(\sim 0.7<x\le 1)$ elastic, thermal, and structural properties. We determine N dilute-point-defect diffusion pathways, activation energies, attempt frequencies, and diffusion coefficients as a function of temperature. In addition, the MD simulations presented in this paper reveal an unanticipated atomistic process, which controls the spontaneous formation of N self-interstitial/N vacancy $({\mathrm{N}}^{\mathrm{I}}/{\mathrm{N}}^{\mathrm{V}})$ pairs (Frenkel pairs), in defect-free TiN. This entails that the N lattice atom leaves its bulk position and bonds to a neighboring N lattice atom. In most cases, Frenkel-pair ${\mathrm{N}}^{\mathrm{I}}$ and ${\mathrm{N}}^{\mathrm{V}}$ recombine within a fraction of ns; ∼50% of these processes result in the exchange of two nitrogen lattice atoms $\left(\mathrm{N}-{\mathrm{N}}^{\mathrm{Exc}}\right)$. Occasionally, however, Frenkel-pair N-interstitial atoms permanently escape from the anion vacancy site, thus producing unpaired ${\mathrm{N}}^{\mathrm{I}}$ and ${\mathrm{N}}^{\mathrm{V}}$ point defects.

Figure 1

N-N interatomic forces as a function of N-N separation distance in ${\mathrm{N}}_2$, as obtained from ab initio and MEAM calculations using the original [47] (5.96) and the present paper's (4.33) $\alpha (\mathrm{N},\mathrm{N})$ MEAM parameter value. The enlarged view around the ${\mathrm{N}}_2$ equilibrium bond length illustrates the results of hybrid functionals.

Figure 2

$B1$ TiN phonon dispersion as calculated in the present work from MEAM using a $\alpha (\mathrm{N},\mathrm{N})$ value of 4.33 in comparison with (a) neutron scattering measurements [76] and (b) DFT [71] results. MEAM results are solid red lines. DFT results are dotted black lines. Open squares and filled circles indicate experimental longitudinal and transversal phonon frequencies, respectively.

Figure 3

$\mathrm{Ti}{\mathrm{N}}_x$ lattice parameters and elastic properties calculated with MEAM (using a MEAM-parameter $\alpha (\mathrm{N},\mathrm{N})=4.33$) as a function of the N concentration $x(0.6<x\le 1)$ in comparison with experimental measurements. $\mathrm{Ti}{\mathrm{N}}_x$ bulk and thin-film elastic constants and moduli determined experimentally: sound velocities by T. Lee et al. [36], neutron scattering by Kress et al. [76], sound velocities by Y. C. Lee et al. [79], sound velocities by Kral et al. [77] and Wolf [78], synchrotron x-ray diffraction by Shebanova et al. [81], Brillouin scattering and nanoindentation by Jiang et al. [80], and nanoindentation by Shin et al. [37]. Ab initio results are taken from previous DFT+PW91 calculations [72], DFT+PW91 by Ashley et al. [73], DFT+LDA by Jhi et al. [74], and DFT+AM05/PBE/LDA (present paper).

Figure 4

Diffusion of N self-interstitials in $B1$ TiN as predicted by AIMD simulations performed at (a) 2000 K and (b), (c), and (d) 3000 K. ${\mathrm{N}}^{\mathrm{I}}$ migrates from one anion site to another by passing through metastable eightfold-coordinated tetrahedral sites (orange dots) located at the center of $B1$ conventional unit cells. ${\mathrm{N}}^{\mathrm{I}}$ diffusion pathways are marked by double-colored solid lines. Black dotted lines evidence the motion of N lattice atoms during ${\mathrm{N}}^{\mathrm{I}}$ migration from one split-bond configuration to another. Gray (blue) balls indicate the equilibrium position of Ti (N) lattice atoms. Atom and diffusion pathway projections onto (100), (010), and (001) box side planes facilitate the visualization of ${\mathrm{N}}^{\mathrm{I}}$ migration trajectories. Each panel represents the dynamics of atoms for a total duration of ∼0.7 ps.

Figure 5

AIMD and CMD N self-interstitial jump rates $ln\left[k^{\mathrm{I}}\right]$ vs inverse temperature $1/T$. CMD runs performed using a MEAM parameter $\alpha (\mathrm{N},\mathrm{N})=4.33$ return jump rates close to the AIMD values. Error bars correspond to uncertainties ${\sigma }_{\kappa }$ (see text). $T_{\mathrm{melt}}\sim 3250\mathrm{K}$ is the melting point of TiN.

Figure 6

AIMD and CMD N self-interstitial diffusion coefficient $D^{\mathrm{I}}$ dependence on temperature. Shaded regions correspond to uncertainty ranges ${\sigma }_D^{\prime }$ estimated as described in Eq. (4).

Figure 7

Diffusion of N vacancies in $B1$ TiN as predicted by AIMD simulations performed at 3000 K. A nitrogen lattice atom migrates from its bulk position into a neighboring vacant anion site by moving along the [101] direction. The ${\mathrm{N}}^{\mathrm{lattice}}$ diffusion pathway is marked by a double-colored solid line. Gray (blue) balls indicate the equilibrium position of Ti (N) lattice atoms. Atom and diffusion pathway projections onto (100), (010), and (001) box side planes facilitate the visualization of the ${\mathrm{N}}^{\mathrm{lattice}}$ migration trajectory to the vacancy site. The figure represents the dynamics of atoms for a total duration of ∼1 ps.

Figure 8

AIMD and CMD N vacancy jump rates $ln\left[k^{\mathrm{V}}\right]$ vs inverse temperature $1/T$. Error bars correspond to uncertainties ${\sigma }_{\kappa }$ (see text).

Figure 9

Schematic representation of Frenkel-pair formation, annihilation, and dissociation. Spontaneous formation of Frenkel pairs in defect-free $B1$ TiN: (a) and (b) a nitrogen lattice atom moves to a neighbor anion site thus forming a ${\mathrm{N}}^{\mathrm{I}}/{\mathrm{N}}^{\mathrm{V}}$ pair. Frenkel pairs are shown in their most stable 0-K configuration: the ${\mathrm{N}}^{\mathrm{I}}/{\mathrm{N}}^{\mathrm{lattice}}$ pair bond axis is oriented along a $\langle 110\rangle$ direction with the two nitrogen $({\mathrm{N}}^{\mathrm{I}}/{\mathrm{N}}^{\mathrm{lattice}})$ atoms equally distant from the vacancy site ${\mathrm{N}}^{\mathrm{V}}$. Frenkel-pair annihilation mechanisms: (c) and (d) the nitrogen lattice atom returns to its initial position; (e) and (f) two nitrogen lattice atoms exchange position in comparison with the original configuration seen in (a). Frenkel-pair dissociation: (g) and (h) one of the two nitrogen atoms forming the ${\mathrm{N}}_2$ split-bond configuration diffuses to an anion site further away from ${\mathrm{N}}^{\mathrm{V}}$.

Figure 10

CMD Frenkel-pair formation rates $\ln [k^{\mathrm{F}}]$ vs inverse temperature $1/T$. Error bars correspond to uncertainties ${\sigma }_k=\pm k^{\mathrm{F}}\left(T\right)/2$ (see text). CMD $\mathrm{N}-{\mathrm{N}}^{\mathrm{Exc}}$ rates are approximately equal to $k^{\mathrm{F}}\left(T\right)/2$. The inset shows AIMD and CMD results at $T>T_{\mathrm{melt}}$.

Figure 11

Estimated ${\mathrm{N}}^{\mathrm{I}}$ and ${\mathrm{N}}^{\mathrm{V}}$ effective contribution to mass transport in bulk TiN between 300 and 1500 K. Upper/lower curves of the same type define upper/lower uncertainty limits.

Figure 12

Calculated (the present paper's CMD and AIMD $D$ values) and experimental nitrogen diffusion coefficients in bulk $B1$ TiN. MD $D^{\mathrm{I}}$ and $D^{\mathrm{V}}$ results correspond to N self-interstitial and N vacancy diffusion coefficients. Experiments 1 and 2 curves are from Ref. [16]. Experiments 3 and $4 D$ results are from Ref. [18]. Experiments 5–8 diffusion coefficients are from Ref. [17].