Statistical Enhancement of Terminal Current Estimation Monte Carlo Device Simulation

We present a new generalized Ramo-Shockley theorem (GRST) to evaluate contact currents, applicable to classical moment-based simulation techniques, as well as semiclassical Monte Carlo and quantum mechanical transport simulation, which remains valid for inhomogeneous media, explicitly accounts for generation/recombination processes, and clearly distinguishes between electron, hole, and displacement current contributions to contact current. We then show how this formalism may be applied to Monte Carlo simulation to obtain equations for minimum-variance estimators of steady-state contact current, making use of information gathered from all particles within the device. Finally, by means of an example, we demon-strate this technique’s performance in acceleration of convergence time.


INTRODUCTION: THE RAMO-SHOCKLEY
THEOREM such that all contacts are grounded with the exception of contact k, which is set to Volt.
Through application of the method of Green functions in the quasi-electrostatic approximation, Shockley and Ramo [2] introduced the original domain integration formula relating the currents induced on an arbitrary number of contacts to the motion of charges in multiple dimensions I (k) -EqjE)'vj, (1) J where the index k indicates the contact at which the current is to be evaluated, qj and vj represent particle charge and velocity, respectively, and the index j runs over all particles within the volume. The symbol E) k) denotes the electric field at the position of particle j which would result if all charges were removed from the volume, and boundary conditions were imposed 303 2 A GENERALIZED RAMO-SHOCKLEY THEOREM One of the most fundamental properties of all types of transport are the continuity equations, which may be derived by taking moments of the appropriate transport equations. For classical and semi-classical systems, this is most often the Boltzmann Transport equation [3], and either the Wigner-Boltzmann equation [4] or the quantum Liouville equation [5] for quantum systems. The 0 th moments take on the following general form: --V'jp eGp(r) + ep, Furthermore, Maxwell's equations may be used to derive equally familiar identities for the displacement current Jd and total current Jr" fahklV "jkdV frh,ljkdS + faVhkl "jkdV.
The definition of hkl requires that the surface integral is equivalent to the contact current, because 0. Equations (2), (3), hkllr 51m and (Jk h)[rN (4), (5), and (7), lead to the following formulas for the terminal currents at an arbitrary contact Inl f vh.( .a.ev + e f ahnt(r)dV e ]Gn(r)hm(r)dV, (8) lpl f Vhpl(r).jpdV e f 1)hpl(r)dV e J Gp(r)hpl(r)dV, This method is also applicable to terminal currents of quantities derived from higher-order moments of the carrier distribution functions, inhomogeneous media and high frequencies [8]. The steady-state form of these equations is obtained by setting all time derivatives equal to zero.

OPTIMIZED STEADY-STATE FORM OF THE GRST
Convergence time for steady-state particle-based simulation is limited by estimator variance. Minimizing the functional form of the steady-state terminal cur-  (16), and used along with simulated carrier density and velocity profiles, such as shown in Fig. 1, to obtain the cumulatively time-averaged steady-state drain currents shown in Fig. 2. All simulations were run with 20000 electrons and 10000 holes. In comparison to the particle counting technique, convergence is reached with the optimzed GRST method between 10 and 40 times faster for these examples.
His professional interests include semiconductor device physics, applied mathematics, and Monte Carlo simulation.
Ulrich Krumbein was born in Braunschweig, Germany in 1964. In 1990 he recieved the Diploma in physics from the University of Kaiserslautern, Germany. In 1991 he joined the Integrated Systems Laboratory at the ETH in Zurich, Switzerland, where he is currcntly working towards his Ph.D. in semiconductor device simulation. His research interests are in physical models for silicon device simulation, with special emphasis on EEPROM devices.
K. Girtner was born in Zittau, Germany, on March 4, 1950. He studied theoretical physics in Dresden (Germany), and received a PhD in nuclear reactor physics. After working ten years in the field of neutron transport, he joined the Karl-WeierstrafS-Institute for Mathematics in Berlin in 1982. Since