Stochastic Model of Breakdown Nucleation under Intense Electric Fields

A plastic response due to dislocation activity under intense electric fields is proposed as a source of breakdown. A model is formulated based on stochastic multiplication and arrest under the stress generated by the field. A critical transition in the dislocation population is suggested as the cause of protrusion formation leading to subsequent arcing. The model is studied using Monte Carlo simulations and theoretical analysis, yielding a simplified dependence of the breakdown rates on the electric field. These agree with experimental observations of field and temperature breakdown dependencies.

Figure 1

Fixed points of the dynamical equations for $\rho$, attracting (${\rho }_{*}$) and repelling (${\rho }_c$), as functions of $E$, demonstrating the existence of a bifurcation point.

Figure 2

Analytical versus simulated values: (a) probability of being in state $n$, at $t\ll \tau$, as calculated from the analytical expression [Eq. (6)] and numerical simulation. The lines, from bottom to top, are for $E=200$, 230, and $260\mathrm{MV}/\mathrm{m}$. The simulation points include measurements of the probability for $n>n_c$, above the QSD validity regime. (b) $\tau$ normalized by $\tau (E_0=180\mathrm{MV}/\mathrm{m})$ as a function of $E$, calculated using the metastable approximation [Eq. (7), solid line], the exact formula [Eq. (5), dashed line], and the simulation (triangles).

Figure 3

Experimental BDRs with fitted theoretical lines using Eq. (7): (a) BDR versus $E$ for various Cu accelerating structures [11]. (b) BDR variation with $E$ at room temperature (two lines on the left) and at 45 K (line on the right) [51]. (c) BDR versus $E$ for various Cu accelerating structures [11, 52], with $E$ rescaled so that all measurements are fitted with $\beta =4.8$.

Figure 4

Fit of free model parameters: (a) LSQ parameter as a function of $\mathrm{\Omega }$ and $E_a$, with $\beta =4.8$ and $\kappa =0.41$. (b) LSQ parameter as a function of $\beta$ and $\kappa$, with $\mathrm{\Omega }=5.4\mathrm{eV}/\mathrm{GPa}$ and $E_a=0.08\mathrm{eV}$.