II Coupled Problems - LMN

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    Technology and Device modeling in micro and nano-electronics:

    current and future challenges

    Andrea Marmiroli, Gianpietro Carnevale, Andrea Ghetti


    Via Olivetti 2, 20041 Agrate Brianza, ITALIA

    Abstract The number of physical effects that have to be

    taken into account to accurately model and design current and II. COUPLED PROBLEMS future micro- and nano-electronics devices is continuously This increased importance of physical mechanisms has a increasing. At the same time, the importance of the coupling clear impact also on the need to better control (and where among them is increasing too. An accurate simulation of such possible exploit) the coupling between different steps. Such effects with strong interactions is often non-trivial and in many couplings can be split in different areas: cases a satisfactory solution is not yet available. Two challenging

    problems are proposed. technological coupling, such as the lithographic and the

     following etching steps have to be considered more and Keywords microelectronics; silicon oxidation; Poisson, more as a unique step; Schrödinger, Boltzmann equations. technological-electrical coupling. It might be argued that

    micro- and nano-electronic are intrinsically a coupling I. INTRODUCTION between technology and electrical performances. Besides

    the obvious and usually controlled interactions, more and The peculiar driving force of micro- and nano-electronic

    more phenomena are affecting the electrical industry is the shrink of dimensions. This shrink allows to

    performances: as mentioned, strain in the devices; but use less silicon and to pack more devices on the same wafer, also the effective dielectric constants of the different reducing the production costs. It is the key factor for the materials; the effective distance between interconnect increase of device performance, thanks to the reduction of lines, the shape of the active area of transistors, the parasitic capacitance and parasitic resistance and to the dimensions and the shape of the floating gates in flash increase of transistor current. It allows the reduction of the memory devices; dimensions of the final equipments (cellular phones, portable electro-thermal coupling: the mutual interactions computers, etc.), increasing the added value of the integrated between power dissipation and electrical performances; circuits. Along this shrink path, the minimum features the new technologies based on Phase Change Materials, defined by today’s technology are in few tens of nanometers for which the transition between “0” and “1” is range [1]. controlled by the temperature generated by Joule effect; Because of this continuous shrink, to design and electromagnetic coupling; including all the Electro manufacture semiconductor devices, more and more physical Magnetic Compatibility and Electro Magnetic Immunity mechanisms, which were previously negligible, have now to issues.

    be taken into account. Among the most important there are:

     diffraction and interference effects in lithography, which In the following, among the long list of problems, two of

    particular interest from a computational point of view, have are posing very challenging computational problems to

    been selected: allow the modeling of the aerial image generated through

    1. the oxidation of silicon in 3 dimensions; complex masks featuring phase shift layers and sub-2. the modeling of transport in today's and future circuits, lithographic patterns; coupling Poisson, Schrödinger and Boltzmann electrical behavior of strained silicon: transistor equations. performance depend on strain fields. This has to be taken

    into account for an accurate modeling and can also be III. SILICON OXIDATION exploited to improve transistor speed; Modeling of silicon oxidation has two main objectives: electro-magnetic coupling between conduction lines first to predict the exact shape of silicon and of silicon oxide, which are closer and closer; then to evaluate the stress and strain in the two films as the resistance of the parasitic interconnect metal lines, which electrical performances, some failure and degradation is now no more negligible; mechanisms are strongly dependant on the stress/strain level effects of power dissipation. [2,3]. An accurate modeling of silicon oxidation has than to

    deal with two linked problems: the diffusion of the oxidizing

    species in silicon oxide and the solution of the mechanical

    problem related to the formation of silicon oxide (whose

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    volume is twice as large as that of the original silicon). These quantum mechanical (QM) and strain effects in the two issues are strongly combined by means of a nonlinear framework of semi-classical 3D MC device simulation. dependency of the main physical quantities which are used in In this paper we report on a new MC simulator (called the diffusion-reaction problem: the diffusivity, the reaction MC++) that solves self-consistently in 1D, 2D or 3D the

    Schrödinger Eq. for the QM correction of the potential, while rate and the oxide viscosity (respectively D, K) and react, oxi

    mechanical strain effects are accounted for by an appropriate the stress quantities which are calculated in the mechanical

    change of the band structure. We will show that QM problem: pressure and maximum shear stress (P, ), as

    corrected 3D semi-classical Monte Carlo device simulation shown in Fig. 1

    can accurately address all the above issues.

    Fig. 2 graphically depicts the interaction among the main

    blocks of MC++. It solves the Schrödinger Eq. (SE) and the

    Poisson Eq. (PE) self-consistently with the semi-classical 3D

    Monte Carlo simulation of carrier transport through an

    iterative procedure.

    The linear PE is solved using standard box methods for

    the potential () profile frequently enough (every 2fs) to

    assure time stability.

     Fig. 1 Solution algorithm coupling oxidation rates and stress calculations Fig. 2. Main blocks of the simulation program and their inter- actions. Simulation starts by reading an initial guess This coupled algorithm has been applied in two computed with conventional programs. dimensions to predict the shape of the isolation oxide in today’s flash memory arrays. The solution of the SE provides the QM correction term The complexity of this task is largely increased by the () of the potential accounting for charge quantization [6]. computational/mathematical problem of managing in three Both and;; act as driving force in the Boltzmann dimensions (3D) the moving boundaries describing the Transport Eq. that is solved for via semi-classical 3D Monte silicon-silicon oxide and the silicon-gas interfaces. Carlo simulation providing carrier/pseudo-potential profiles

    to be used in the solution of both PE and SE. IV. POISSON-SCHRÖDINGER -BOLTZMANN By coupling self-consistently the three blocks of Fig. 2 it The second case deals with device simulation of silicon is possible to achieve good agreement with experimental data NanoWire MOSFETs. NanoWire MOSFETs like the one [6]. reported in [4] are gaining increasingly popularity due to

    their superior channel control. This is achieved by reducing V. CONCLUSIONS the silicon channel to a thin wire surrounded as much as The continuous technology shrinking mandates the need possible by the gate. This makes this kind of devices to account for more and more coupled physical effects. We intrinsically 3D. showed how this is the case for two of the most important In addition, highly non-equilibrium transport still problems in the technology and device modeling area. These dominated by scattering is expected in this kind of devices [5]. developments require further investment and collaboration This complex non-stationary/ballistic transport can be between industry, research center and software vendors, in accurately accounted for by semi-classical Monte Carlo order to provide accurate tools in time for effective usage. simulation. However, for such small devices, quantum

    mechanical and strain-induced effects play a fundamental role

    that must be accounted for in conjunction with the real 3D

    geometry of the device. Therefore it is necessary to include

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    We wish to thank A. Benvenuti, L. Baldi, L. Vendrame, P. Fantini for their different relevant contributions.


    [1] International Technology Roadmap for Semiconductors 2005, . [2] K. Rim et al., Symp. VLSI Tech., 2001 [3] S. Thompson et al., IEEE Trans. Elec. Dev., vol 51, n. 11, p. 1790,2004 [4] J. Wang et al, J. of Appl. Phys., vol. 96, n. 4, p. 2192, 2004 [5] MJ. Gilbert et al, J. of Appl. Phys., vol. 98, n. 9, 094303, 2005 [6] A. Ghetti et al, to be presented to SISPAD’06

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