Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

4.6 Conclusion

A stable backward method has been developed and implemented to overcome the statistical drawbacks of the forward method. The method allows one to calculate the current in the entire sub-threshold region including the leakage current in the off-state. Symmetric current estimators are proposed which produce less statistical error than the non-symmetric ones. This improvement is achieved for all operating conditions and is particularly large when thermal equilibrium is approached [P4].

The current through a plane is calculated by Monte Carlo integration of the current density. For this integration, one has to assume a distribution of the sampling points which in the present case are the initial wave vectors of the backward trajectories. By assuming a Maxwellian distribution at elevated temperature, the method will generate more sampling points at higher energies. This method of statistical enhancement reduces the statistical error of quantities that depend on the high-energy tail of the distribution function. It is shown that the estimated current is independent of the injection temperature, whereas the statistical error shows a clear minimum where the injection distribution most closely resembles the actual distribution [P4].

The proposed backward Monte Carlo method is able to estimate the energy distribution function in a chosen point in the (, ) phase space with the desired accuracy. The high-energy tail of the distribution can be calculated point-wise.