Fick's law of diffusion was used to model the diffusion of Cu into Ta barrier material. By matching experimental results, the Cu diffusion coefficient was found to be described by:

D(cm2/s) = 2.870 x 10−14exp[−0.1457(eV)/kT]cm2/s

By using the calculated results, a 25nm Ta layer was found to be sufficient to stop Cu from diffusing through it during annealing (600C, 0.5h).

Study of Copper Diffusion into Ta and TaN Barrier Materials for MOS Devices. S.W.Loh, D.H.Zhang, C.Y.Li, R.Liu, A.T.S.Wee: Thin Solid Films, 2004, 462-463, 240-4