Dielectric breakdown of TMR sensors and the role of Joule heating

For voltage (V) stress of TMRs, the log of the dielectric breakdown time is found to be linear in (H-γZaV/t_B)/(k_BT), where H, γ, Z, a, t_B and k_B are an activation energy, a parameter, charge, tunnel barrier lattice constant and thickness, and Boltzmann constant. T is the ambient plus Joule heating temperature of the tunnel barrier.