The Physics of Non–Equilibrium Reliability Phenomena

Chapter E Calculation of Bond Breaking Rates

The Si–H bond breaking rate ultimately determines the created damage at the Si/SiO\(_2\) interface and, hence, affects the simulated reliability characteristics of device, rendering the calculation of \(\Gamma ^\mathrm {break}\) of particular importance. Below, different concepts to calculate \(\Gamma ^\mathrm {break}\) are introduced and compared against each other, eventually motivating the choice of using the WKB

First, a purely classical rate approach is given by

\begin{equation} \Gamma _\mathrm {cla}^\mathrm {break}=\sum _i P_i \Gamma _{i,\mathrm {cont}}^\mathrm {tot}, \end{equation}

with \(P_i\) being the quasi–equilibrium population of state \(i\) and \(\Gamma _{i,\mathrm {cont}}\) is the excitation rate from state \(i\) to the first level above the transition barrier between the left and right well, referred to as continuum state. The classical rate only takes into account the left well of the ground state potential.

Second, a semi-classical calculation of the breakage rates is given by the WKB approximation

\begin{equation} \begin{aligned} &\Gamma _\mathrm {WKB}^\mathrm {break}=\sum \limits _{i=1}^{f}\Gamma _{i,\mathrm {WKB}}\,P_{i}=\\ &=\sum \limits _{i=1}^{f}\exp \bigl (-\frac {2}{\hbar }\int \limits _{x_{1,i}}^{x_{2,i}}\sqrt {2(V-E_i)} \mathrm {d}x\bigr )P_i, \end {aligned} \end{equation}

where \(P_i\) is again the state population and \(\Gamma _{i,\mathrm {WKB}}\) is given by the tunneling probability through the barrier between the classical turning points \(x_1\) and \(x_2\). Only eigenstates in the left well which have a partner of similar energy in the right well are included in the calculation¹.

Finally, propagating the density matrix in time is the most rigorous approach and allows one to directly access the population dynamics of the system. Starting with a 100% localization in the left well of the ground state, e.g. \(P_0=1,\,P_{i\ne 0}=0\), the transition rate can be defined as

\begin{equation} \Gamma _\mathrm {DM}^\mathrm {break}=-\frac {\mathrm {d}P_\mathrm {L}(t)}{\mathrm {d}t}=\frac {\mathrm {d}P_\mathrm {R}(t)}{\mathrm {d}t}, \end{equation}

assuming a negligible back flow of population. \(P_\mathrm {L,R}(t)\) are the total populations of the left and right well, respectively, at time \(t\). The calculations presented here use a Runge–Kutta integrator of fourth order with a timestep of 100 a.u. and a total propagation time of \(\SI {10}{ns}\).

The results for all three different rates at selected points along the Si/SiO\(_2\) interface are summarized in Fig. E.1 and Table E.1. One can see that the classical approach \(\Gamma _\mathrm {cla}^\mathrm {break}\) underestimates the rate by at least one order of magnitude, whereas the WKB method and the density matrix propagation give similar values. Although \(\Gamma _\mathrm {DM}^\mathrm {break}\) employs the most accurate method, its computational effort is unfeasible for the work presented here. Taking into account that the numerical simulation of a MOSFET and the energy distribution function includes several hundred mesh points at the interface as well as the inherent distribution of parameters, which requires to randomly sample the parameter space, would have made the calculations prohibitively expensive.

	\(\Gamma ^\mathrm {break}_\mathrm {Source}\) [\(\SI {}{s^{-1}}\)]	\(\Gamma ^\mathrm {break}_\mathrm {Channel}\) [\(\SI {}{s^{-1}}\)]	\(\Gamma ^\mathrm {break}_\mathrm {Drain}\) [\(\SI {}{s^{-1}}\)]
classical	\(\SI {1.6e-18}{}\)	\(\SI {3.6e-7}{}\)	\(\SI {3.8e2}{}\)
semi–classical WKB	\(\SI {2.4e-16}{}\)	\(\SI {1.0e-6}{}\)	\(\SI {2.1e3}{}\)
\(\rho \) propagation (\(\frac {\mathrm {d}P_\mathrm {tot,R}}{\mathrm {d}t}\))	\(\SI {6.2e-16}{}\)	\(\SI {1.1e-6}{}\)	\(\SI {4.8e3}{}\)

Table E.1: The calculated bond breaking rates \(\Gamma ^\mathrm {break}\) using the three different approaches, classical, Wkb and propagation the density matrix. While the latter to variants yield very similar rates, the classical calculation method tends to consistently underestimate \(\Gamma ^\mathrm {break}\).

¹ The WKB method, at a first glance, seems physically incorrect due to the orthogonality of the wavefunctions in the left and right well, respectively. However, the resonance mediated process via an intermediate potential effectively enables the coupling of \(\phi _\mathrm {L}\) and \(\phi _\mathrm {R}\).