Stochastic Properties of Monte-Carlo Device Simulations
The relative stochastic error of a quantity evaluated by stationary MC simulation is proportional to its variance, which can be evaluated with the corresponding autocorrelation function (cf. Sec. 3.4).
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Stochastic Properties of Monte-Carlo Device Simulations
The relative stochastic error of a quantity evaluated by stationary MC simulation is proportional to its variance, which can be evaluated with the corresponding autocorrelation function (cf. Sec. 3.4). Since the spectral intensity is the Fourier transform of the autocorrelation function (2.66), the variance of a quantity X averaged over the simulation time Tsim (Eq. (3.60)) can be evaluated with the spectral intensity similar to :\lac Donal(i.' s function
J (1 - kL) J T;oim
o-2{X} Tsim
C{X}(T)dT
T slln
-Tsim
:x:
_1_ 7rTsim
o 5 x .1«(0)
5 yy (~) 1 - cos(v) dv .. T sim v2
2Tsim
(8.1) (8.2)
where the approximation (8.2) holds for sufficiently large simulation times. Without statistical enhancement the spectral intensity can be calculated exactly with the methods of Sec. 2.4 or approximately with the methods of Chap. 7. The latter approach is used here in a modified form to investigate in Sec. 8.1 the properties of terminal current estimators. In Sec. 8.2 it is shown how to estimate the CPU time of MC simulations in-advance with the momentum-based noise models of Chap. 7.
C. Jungemann et al., Hierarchical Device Simulation © Springer-Verlag Wien 2003
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8.1 Stochastic Error
801
Stochastic Properties of Terminal Current Estimators
In this section the stochastic noise of stationary MC device simulations is discussed for a generalization of the Ramo-Shockley estimator for terminal currents. Under stationary bias conditions the expected value of the displacement current density within the device is zero and Eq. (6.74) reduces to [8.1] (8.3) The stochastic noise of this estimator is characterized by the spectral intensity of its fluctuations (cf. Eq. (8.1)), which can be calculated with sufficient precision with the methods described in Chap. 7. Only in the case of very high frequencies (> 100GHz) a modified approach must be used, which includes the first order derivatives with respect to time in Eqs. (7.17)-(7.30). But as already pointed out in Chap. 7 this causes numerical problems. To circumvent these problems in the following the DD approach including all first order derivate with respect to time is used only in the case of equilibrium for which these instabilities do not occur. The linearized Langevin-type DD model in the frequency domain is derived in this case by calculating balance equations for the microscopic quantities [8.2]
x x
1,
(8.4)
TV
(8.5)
1 + iWT
avoiding the macroscopic relaxation time approximation (cf. Sec. 2.4). The resultant small-signal DD model for the fluctuations reads under equilibrium conditions (V'r
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