3.8.2.2 Capture and Emission Times

Once the capture and emission probabilities have been obtained, the corresponding times can be calculated. The inverse of the capture time is given by [219,223]

$$\ensuremath{\tau_\mathrm{c}}^{-1}(x) = \int_{\ensuremath{{\mathca... ...suremath{f_\mathrm{c}}({\mathcal{E}})\,\ensuremath {\mathrm{d}}{\mathcal{E}}\ ,$$ (3.145)

where $\ensuremath{g_\mathrm{c}}({\mathcal{E}})$ denotes the two-dimensional density of states and $\ensuremath{f_\mathrm{c}}({\mathcal{E}})$ the electron energy distribution function in the cathode. For the above stated assumption that all electrons are captured from the same energy level $\ensuremath {{\mathcal{E}}_\mathrm{c}}+3/2k_{B}T$ in the cathode, this expression can be approximated by

$$\tau _{c}^{-1}(x) \approx \ensuremath{W_\mathrm{c}}\left(x,\ensur... ...prime}},\ensuremath {{\mathcal{E}}_\mathrm{c}}+\frac{3}{2}k_{B}T\right)n_{c}\ ,$$ (3.146)

where $n_{c}$ is the sheet carrier concentration in the cathode, which is determined by the transport model used in the device simulator. The inverse of the emission time is [219]

$$\ensuremath{\tau_\mathrm{e}}^{-1}(x) = \int_{-\infty}^{\ensuremat... ...h{f_\mathrm{a}}({\mathcal{E}})\right)\,\ensuremath {\mathrm{d}}{\mathcal{E}}\ .$$ (3.147)

Assuming $\ensuremath{f_\mathrm{a}}({\mathcal{E}}) \approx 0$ in the anode and elastic tunneling for the emission process ( ${\mathcal{E}}= \ensuremath{{\mathcal{E}}^{\prime}}$), the emission time becomes

$$\ensuremath{\tau_\mathrm{e}}^{-1}(x) \approx \ensuremath{W_\mathr... ...{g_\mathrm{a}}(\ensuremath{{\mathcal{E}}^{\prime}}) \ensuremath{\hbar\omega}\ ,$$ (3.148)

where the energy loss is restricted to values less than $\ensuremath{\hbar\omega}$. To check the validity of the approximations for the wave functions, the resulting capture and emission times have been compared to results using a SCHRÖDINGER-POISSON solver for a MOS capacitor with the parameters ${\mathcal{E}}_\mathrm{T}$=2.8eV, $S\hbar\omega$=1.6 eV, and a trap concentration of $\ensuremath{N_\mathrm{T}}=1e19 cm-3$. As can be seen in Fig. 3.19, the analytical and the numerical results are very close. Electrons are captured from the right and emitted to the left in this figure. Thus, for traps near the right side of the barrier the capture time is very low and the emission time is very high. The oscillations in the emission time for high bias are due to the fact that in this regime, the energy barrier has a triangular shape which gives rise to an oscillating wave function, in contrast to the decaying wave function for a trapezoidal barrier.

Figure 3.19: Comparison of the analytic solution with a numerical solution for the capture and emission times at a gate bias of 3 V (left) and 7 V (right).

A. Gehring: Simulation of Tunneling in Semiconductor Devices