Ab initio study of Cu impurity diffusion in bulk TiN

TiN is an important material used as a diffusion barrier in microelectronic devices to prevent copper from contacting silicon. There is, however, little known about the elementary atomistic processes underlying the excellent performance of TiN. In this work, we perform a density functional theory study of the copper impurity diffusion coefficient in bulk TiN. Several diffusion mechanisms are considered. For each mechanism, the temperature effect is taken into account within the quasiharmonic approximation. Moreover, the influence of the TiN stoichiometry on its diffusion properties is taken into account through the change in the concentrations of the intrinsic point defects as a function of composition. These concentrations are obtained via a thermodynamic formalism based on the dilute solution model. We find that in stoichiometric TiN the copper impurity diffusion proceeds via the vacancy mechanism on the Ti sublattice. In the off-stoichiometric ${\mathrm{TiN}}_{0.96}$, the dominant diffusion mechanism switches to the vacancy-mediated diffusion on the N sublattice. The Arrhenius equation for the diffusion coefficient is $D=3.8\times {10}^{-4}exp(-4.5\mathrm{eV}/k_{B}T){\mathrm{m}}^2{\mathrm{s}}^{-1}$ for the stoichiometric TiN and $D=6.7\times {10}^{-8}exp(-2.7\mathrm{eV}/k_{B}T){\mathrm{m}}^2{\mathrm{s}}^{-1}$ for the substoichiometric TiN. Our calculations provide the basis for a better interpretation of the experimental measurements.

Figure 1

Schematic representation of the “five-frequency” model in fcc lattice (see text). Gray and white balls are the solvent atoms in the different (100) planes, black ball is the impurity atom, and the squares illustrate the vacancies.

Figure 2

(a) Lattice parameter $a$, (b) thermal expansion coefficient $\alpha$, and (c) heat capacity $C_p$ per formula unit of TiN as a function of temperature calculated in this work (full lines), compared to the available experimental [47, 48, 49, 50, 51, 52, 53, 54] (dots) and theoretical [55, 56, 57, 58] (dashed lines) data.

Figure 3

(a) Concentrations of vacancies in TiN as a function of composition calculated for the various temperatures. (b) The formation free energy $G_f$ of the ${\mathrm{Cu}}_{\mathrm{X}}$ impurity atom on the different sublattices X (Ti, N, I) in TiN. The energies are shown relative to the one of ${\mathrm{Cu}}_{\mathrm{N}}$. The full and the dashed lines correspond to the temperatures of 650 K and 1250 K, respectively.

Figure 4

Minimum energy path during migration of N atom in the vicinity of the copper impurity together with the structures of the intermediate images at 1250 K. 1 denotes the migrating nitrogen atom, 2 is the copper impurity atom, 3 is the initial position of nitrogen atom 1, and 4 is the initial position of the nitrogen vacancy where the atom 1 is migrating to.

Figure 5

Arrhenius plot of the calculated diffusion coefficient together with the experimental data. The red line with filled circles corresponds to the calculated diffusion coefficient in N-deficient ${\mathrm{TiN}}_{0.96}$, and the blue one with open circles corresponds to stoichiometric TiN. The green diamond represents the measurement for single-crystalline TiN [22], while other experimental data are obtained from the measurements in polycrystalline TiN [16, 17, 18, 19, 23]. An additional dashed green line represents a linear extrapolation of the experimental data [18, 19] obtained at temperatures above $800{}^{\circ }\mathrm{C}$ which we refer to as “bulk diffusion,” related to the calculated diffusion coefficient in N-deficient ${\mathrm{TiN}}_{0.96}$ (shown for illustrative purposes; see the text).