3.8.2.4 Transient Current

Models of trap-assisted transitions are commonly employed to calculate steady-state SILC in MOS capacitors, while transient SILC has hardly been studied [194,205]. However, transient tunneling current becomes important at high switching speed where the transients of the trap charging and discharging processes may degrade signal integrity. For the calculation of transient SILC it is necessary to calculate capture and emission times at each time step. Considering a spatial trap distribution $\ensuremath{N_\mathrm{T}}(x)$ across the dielectric layer, the rate equation for the concentration of occupied traps at position $x$ reads

$$\ensuremath{N_\mathrm{T}}(x)\displaystyle\frac{\ensuremath {\math... ...m{T}}(x)\ensuremath{f_\mathrm{T}}(x,t)\ensuremath{\tau_\mathrm{e}}^{-1}(x,t)\ ,$$ (3.150)

where $\ensuremath{f_\mathrm{T}}(x,t)$ is the trap occupancy function and $\ensuremath{\tau_\mathrm{c}}(x,t)$ and $\ensuremath{\tau_\mathrm{e}}(x,t)$ are the inverse capture and emission times of electrons by a trap placed at position $x$. In the static case capture and emission processes are in equilibrium and $\ensuremath {\mathrm{d}}\ensuremath{f_\mathrm{T}}(x,t)/\ensuremath {\mathrm{d}}t = 0$. In the transient case, however, capture and emission times include transitions from the cathode and the anode (compare Section 3.8.1.2 and Fig. 3.16)

$$\begin{array}{l} \ensuremath{\tau_\mathrm{c}}^{-1}(x,t) = \en... ...-1}(x,t)+\ensuremath{\tau_\mathrm{ec}}^{-1}(x,t)\ , \end{array}$$ (3.151)

where $\ensuremath{\tau_\mathrm{ca}}$ and $\ensuremath{\tau_\mathrm{cc}}$ are the capture times to the anode and to the cathode, and $\ensuremath{\tau_\mathrm{ea}}$ and $\ensuremath{\tau_\mathrm{ec}}$ the corresponding emission times. To calculate the local trap occupancy, the differential equation (3.150) must be solved. If the capture and emission times $\ensuremath{\tau_\mathrm{c}}^{-1}$ and $\ensuremath{\tau_\mathrm{e}}^{-1}$ are constant over time, like in a discharging process with a constant potential distribution, the solution of (3.150) can be given in a closed form

$$\ensuremath{f_\mathrm{T}}(x,t) = \ensuremath{f_\mathrm{T}}(x,0)\e... ...( 1-\exp \left( -\frac{t}{\ensuremath{\tau_\mathrm{m}}(x,t)}\right) \right) \ ,$$ (3.152)

with $\ensuremath{\tau_\mathrm{m}}^{-1}=\ensuremath{\tau_\mathrm{c}}^{-1}+\ensuremath{\tau_\mathrm{e}}^{-1}$.

A more general approach is to look at the change of the trap distribution at discrete time steps. Integration of (3.150) in time between $t_{i}$ and $t_{i+1}$ and changing to discrete time steps yields

$$\ensuremath{f_\mathrm{T}}(x, t_i)-\ensuremath{f_\mathrm{T}}(x, t_... ..._i - \ensuremath{\tau_\mathrm{m}}^{-1}(x, t_{i-1}) \overline{f_i}\Delta t_i \ ,$$

where the abbreviations $\Delta t_i = t_i-t_{i-1}$ and $\overline{f_i} = \left(\ensuremath{f_\mathrm{T}}(x, t_i) + \ensuremath{f_\mathrm{T}}(x, t_{i-1})\right) / 2$ have been used. Thus it is possible to write the trap distribution over time in the following recursive manner:

$$\ensuremath{f_\mathrm{T}}(x,t_i) = A_i + B_i \ensuremath{f_\mathrm{T}}(x,t_{i-1}) \ ,$$ (3.153)

where the symbols $A_i$, $B_i$, and $C_i$ are calculated from

$$\renewedcommand{arraystretch}{2.2}\begin{array}{l} A_{i} = \displ... ... \frac{\ensuremath{\tau_\mathrm{m}}^{-1}(x, t_i) \Delta t_i }{2}\ . \end{array}$$ (3.154)

Once the time-dependent occupancy function in the dielectric is known, the tunnel current through each of the interfaces is

$$\ensuremath{J_\mathrm{TAT, Anode}}(t) = \ensuremath {\mathrm{q}}\int_{0}^{\ensuremath{t_\mathrm{diel}}}\e... ...ath{\tau_\mathrm{ea}}^{-1}(x, t) \right) \right) \,\ensuremath {\mathrm{d}}x\ ,$$ (3.155)

$$\ensuremath{J_\mathrm{TAT, Cathode}}(t) = \ensuremath {\mathrm{q}}\int_{0}^{\ensuremath{t_\mathrm{diel}}}\e... ...ath{\tau_\mathrm{ec}}^{-1}(x, t) \right) \right) \,\ensuremath {\mathrm{d}}x\ .$$ (3.156)

A. Gehring: Simulation of Tunneling in Semiconductor Devices