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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

4.5 Results and Discussion

The BMC method, as well as the different estimators described above, have been implemented in the full-band Monte Carlo simulator VMC [P1]. Backward trajectories are constructed in the same manner as forward trajectories. Routines for the computation of the free flight and the after scattering states can be used without modification.

For testing purposes the structure of a planar n-channel MOSFET with a gate length of \( L_G = \SI {65}{nm} \), an effective oxide thickness of \( t_\mathrm {ox} = \)2.5 nm, and a channel width of \( W=\SI {1}{\mu m} \) is used. Device geometry and doping profiles have been obtained by process simulation [104]. A sketch of the device structure is shown in Fig. 4.6. Room temperature is assumed for all simulations (\( T_D=\SI {300}{K} \)). The following results and discussions are found in a previous work [P4].

(-tikz- diagram)

Figure 4.6: Sketch of a MOSFET.

4.5.1 Transfer Characteristics

The transfer characteristics have been calculated using the classical device simulator Minimos-NT [71, 94], the conventional FMC method, and the novel BMC method. Each bias point is calculated with \( 10^6 \) trajectories, both with the backward and forward methods.The maximum of the energy barrier determines the location of the injection plane. It is located at \( x_0 = \SI {10.2}{nm} \) relative to the left edge of the gate contact. Fig. 4.7 shows the transfer characteristics. Good agreement between the classical device simulation and the MC simulations is found. The BMC method works well in the entire sub-threshold region, whereas the FMC method (without statistical enhancement) can cover only a few orders of magnitude of the current. The barrier height in the channel increases with decreasing gate voltage. Thus, at some point none of the forward trajectories will be able to surmount the barrier, giving an estimated current of \( I=0 \).

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Figure 4.7: Transfer characteristics of the nMOSFET for two drain voltages, simulated with Minimos-NT, the backward and the forward MC methods.

Further, the statistical error of the BMC method is depicted in Fig. 4.8. In a MOSFET the current component due to carriers injected at the source contact is nearly independent of the drain voltage, whereas the current component of carries originating from the drain contact depends strongly on the drain voltage. At \( V_{DS}=\SI {2.2}{V} \) the back diffusion current from the drain is extremely small, and the total current is dominated by forward diffusion, which will result in a low variance. At \( V_{DS}=\SI {50}{mV} \), on the other hand, the back diffusion current is significant, and a stronger compensation of the two current components takes place, which will result in a higher variance. This explanation, using the forward time picture also holds true in the backward time picture. There a large difference in the two current components is reflected by a significant difference in the statistical weights of the forward and backward diffusing carriers.

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Figure 4.8: Relative Error (relative standard deviation) of the drain current for two drain voltages. Each bias point is calculated with \( 10^6 \) backward trajectories. The current estimator (4.37) was used.

In Fig. 4.9 the computation times for a given error tolerance of \( 10^{-2} \) are compared. In the on-state (\( V_{GS}= \SI {2.2}{V} \)) BMC is about five times faster than FMC. Although in this operating point the energy barrier in the channel is almost completely suppressed, many electrons injected at the source contact get reflected by the geometrical constriction at the source-channel junction. Since the BMC method needs not simulate these reflected carriers, it shows a clear gain also in the on-state. The last point that could be simulated with FMC within a reasonable time was \( V_{GS}= \SI {0.8}{V} \). In this operating point, BMC is about 2300 times faster than FMC as shown in Fig. 4.9.

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Figure 4.9: Computation times by a single core of an Intel i7 processor. The operating points of the transfer characteristics at \( V_{DS}=\SI {2.2}{V} \) are considered. A relative standard deviation of \( 10^{-2} \) is assumed.

4.5.2 Output Characteristics

Fig. 4.10 compares the output characteristics computed by three different methods. As shown in Fig. 4.11, the statistical error decreases with increasing \( V_{DS} \), a trend already discussed in the previous section. The figure also shows that the variance of the symmetric estimator (4.56) is lower in the entire range of drain voltages. Especially at low \( V_{DS} \), where the device is approaching thermal equilibrium, the variance of the non-symmetric estimator tends to explode, whereas the variance of the symmetric estimator shows only a slight increase. In this regime, variance reduction by the symmetric estimator is particularly effective.

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Figure 4.10: Output characteristics of the MOSFET for two gate voltages, simulated with MinimosNT, the forward and the backward MC methods.

Evaluation of the symmetric estimator (4.56) requires the computation of two numerical trajectories. To obtain a fair comparison of the two estimators at equal computational cost, we compute \( N=10^6 \) realizations of the non-symmetric estimator (4.39) and only \( N=5\cdot 10^5 \) realizations of the symmetric estimator. Despite the sample size being smaller in the latter case, this smaller sample gives the lower statistical error.

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Figure 4.11: Relative errors of the output characteristics at two gate voltages. The non-symmetric and the symmetric estimator based on the velocity-weighted Maxwellian are compared. Each bias point involves the calculation of \( 10^6 \) backward trajectories.

4.5.3 Injection from a Non-equilbrium Distribution

The injection distribution \( f_0 \) can be freely chosen and does not have any influence on the expectation value, but it does affect the estimator’s variance. This fact is demonstrated by generating the random states \( \kv _0 \) from non-equilibrium Maxwellian distributions at different temperatures. The operating point is \( V_{GS}=\SI {0.6}{\volt } \) and \( V_{DS}=\SI {2.2}{\volt } \). The current is calculated using (4.43) in conjunction with the estimators (4.37) and (4.39). Fig. 4.12 shows the independence of the estimated current from the injection temperature \( T_0 \).

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Figure 4.12: Temperature stability of the current estimators (4.37), (4.39) and (4.56) which are based, respectively, on a Maxwellian (MW) and a velocity-weighted Maxwellian (vel.weighted MW) injection PDF. Operating point is \( V_{GS}=\SI {0.6}{\volt } \) and \( V_{DS}=\SI {2.2}{\volt } \).

The estimators’ relative errors are compared in Fig. 4.13. Below 700 K, estimator (4.39) shows less statistical error than estimator (4.37). For both estimators, the relative error shows a clear minimum, which can be explained as follows: the more the injection PDF \( f_0 \) resembles the real flux density \( v_x f \), the lower is the current estimator’s variance. From Fig. 4.13 one can conclude that a velocity-weighted Maxwellian at 290 K is the best approximation of the real flux \( v_x f \) at the injection plane. With increasing and decreasing \( T_0 \) the difference between \( f_0 \) and the real flux \( v_x f \) becomes larger and the relative error increases. For \( T_0 > \SI {700}{\kelvin } \) the velocity-weighted Maxwellian is a worse approximation to the real flux \( v_x f \) than the Maxwellian and thus shows a higher variance.

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Figure 4.13: Comparison of the relative errors of the current estimators (4.37), (4.39) and (4.56). Operating point is \( V_{GS}=\SI {0.6}{\volt } \) and \( V_{DS}=\SI {2.2}{\volt } \).

4.5.4 Energy Distribution Function

Figure 4.14 shows the energy distribution function (EDF) for full-bands at three surface points in the channel of the MOSFET. The forward MC simulation performed with \( 10^9 \) trajectories can resolve only a few orders of magnitude of the EDF. Then again, with the backward MC method, the EDF is calculated point-wise with \( 10^4 \) trajectories per point using the estimator (4.23). The EDF shows a Maxwellian tail. One can compute as many orders of magnitude of the high energy tail as needed.

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Figure 4.14: Energy distribution function for full-bands at three surface points in the channel. The distances from the left edge of the gate electrode are given. Operating point is \( V_{GS}=\SI {2.2}{\volt } \) and \( V_{DS}=\SI {2.2}{\volt } \). Solid lines: Forward MC simulation with \( 10^9 \) trajectories. Dotted lines: High energy tails computed with the backward MC method.

4.5.5 Hot Carrier Degradation

In long channel devices and high-voltage MOSFETs degradation is triggered by hot carriers. It is assumed that degradation is caused by the breaking of Si-H bonds at the silicon-oxynitride/silicon interface [104]. The bond dissociation rates are modeled by the acceleration integral, which has the general form [69]

(4.71) \begin{equation} I_A = \sigma _0 \int \limits _{E_{\mathrm {th}}}^\infty \left (E - E_\mathrm {th}\right )^p\,v(E)\, f(E)\,g(E)\,\d E\,. \label {eq:accel-integral-E} \end{equation}

\( E_\mathrm {th} \) denotes an energy threshold, \( g(E) \) the density of states, and \( v(E) \) the group velocity. For the purpose of MC estimation, (4.71) is converted into a \( \kv \)-space integral.

(4.72) \begin{equation} I_A = \sigma _0 \int \limits _{\mathrm {BZ}} \Theta \left (\Epsilon (\kv ) - E_\mathrm {th}\right )\, \left (\Epsilon (\kv ) - E_\mathrm {th}\right )^p\, \left |\vv (\kv )\right |\, f(\kv )\,\dk \end{equation}

Here, \( \Theta \) is the unit step function. For the process considered here in which one hot carrier is able to break a bond, an exponent of \( p=11 \) and an energy threshold of \( E_\mathrm {th} = \SI {1.5}{eV} \) are assumed [104].

We used the combined backward/forward MC method (Section 4.4) to evaluate the acceleration integral. The statistical average is calculated from the forward trajectories using the before-scattering method [46]. In this simulation, \( 10^{10} \) scattering events have been computed. To enhance the number of numerical trajectories at high energies the injection temperature \( T_0 \) has been raised significantly (5000 K and 10 000 K).

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Figure 4.15: Acceleration integral for a 65nm nMOS, simulated with FMC and BMC method at different injection temperatures.

In Fig. 4.15 the MC results are compared to the result of ViennaSHE, a deterministic solver for the BTE based on a spherical harmonics expansion of the distribution function [108]. Fig. 4.15 shows that the MC results are independent of the injection temperature. ViennaSHE predicts higher values in the first part of the channel where carrier heating is still moderate. One could attribute this difference to the band structure model which is more approximate in ViennaSHE than it is in VMC.

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