Stochastic Monte Carlo Technique Research Articles

Optimal Sample Weights for Hemispherical Integral Quadratures

AbstractThis paper proposes optimal quadrature rules over the hemisphere for the shading integral. We leverage recent work regarding the theory of quadrature rules over the sphere in order to derive a new theoretical framework for the general case of hemispherical quadrature error analysis. We then apply our framework to the case of the shading integral. We show that our quadrature error theory can be used to derive optimal sample weights (OSW) which account for both the features of the sampling pattern and the bidirectional reflectance distribution function (BRDF). Our method significantly outperforms familiar Quasi Monte Carlo (QMC) and stochastic Monte Carlo techniques. Our results show that the OSW are very effective in compensating for possible irregularities in the sample distribution. This allows, for example, to significantly exceed the regular convergence rate of stochastic Monte Carlo while keeping the exact same sample sets. Another important benefit of our method is that OSW can be applied whatever the sampling points distribution: the sample distribution need not follow a probability density function, which makes our technique much more flexible than QMC or stochastic Monte Carlo solutions. In particular, our theoretical framework allows to easily combine point sets derived from different sampling strategies (e.g. targeted to diffuse and glossy BRDF). In this context, our rendering results show that our approach overcomes MIS (Multiple Importance Sampling) techniques.

Stochastic simulation of thin photoresist film dissolution: a dynamic and a quasi-static dissolution algorithm for the simulation of surface and line-edge roughness formation

Stochastic simulation techniques gain increased importance in the field of microelectronic processes especially since the sub-100 nm device dimensions have been reached. In this work, stochastic Monte Carlo techniques are incorporated to simulate photoresist polymer chains in a lattice, while all the phenomena taking place during film dissolution are considered in terms of occurrence probabilities in order to describe the dynamics of dissolution. Chain removal is based on the critical ionization fraction criterion. The exponential decrease of dissolution rate with increasing polymerization length is proven and its relation to critical ionization is investigated both in two and three dimensions (2D/3D), resulting in an efficient method for the determination of the dissolution rate in terms of polymerization length and critical ionization fraction. While the dynamic dissolution algorithm is appropriate for obtaining information about dissolution rate and surface roughness evolution, the increased computational time in high values of critical ionization fraction and lattice sizes, especially in 3D, make it inappropriate for line-edge roughness studies. A quasi-static 2D/3D resist dissolution algorithm, which is free of the dissolution blocking problems and orders of magnitude faster than the dynamic one, based again on the critical ionization criterion, is constructed in order to reliably quantify line-edge roughness.

Monte Carlo simulation of coherent effects in multiple scattering

Using a combination of the stochastic Monte Carlo technique and the iteration procedure of the solution to the Bethe–Salpeter equation, it has been shown that the simulation of the optical path of a photon packet undergoing an n th scattering event directly corresponds to the n th–order ladder diagram contribution. In this paper, the Monte Carlo technique is generalized for the simulation of the coherent back–scattering and temporal correlation function of optical radiation scattered within the randomly inhomogeneous turbid medium. The results of simulation demonstrate a good agreement with the diffusing wave theory and experimental results.

Coherent Effects in Multiple Scattering of Linearly Polarized Light

Comparing the stochastic Monte Carlo technique with the iteration procedure for solving the Bethe-Salpeter equation in the framework of numerical simulation, the time correlation function and the interference component of the coherent backscattering of a linearly polarized light wave in a multiply scattering medium are calculated. The results of the simulation agree well with theoretical results obtained by generalizing the Milne solution, as well as with experimental data.

Numerical simulation of coherent effects under conditions of multiple scattering

The time correlation function and the interference component of the coherent backscattering from a multiple-scattering medium are calculated in the framework of the Monte Carlo technique. By comparing the stochastic Monte Carlo technique with the iteration procedure of solving the Bethe-Salpeter equation, it is shown that the simulation of the optical path of photon packets that have experienced n scattering events is exactly equivalent to calculating the nth-order ladder diagram. Using this equivalence, the Monte Carlo technique is generalized for simulation of the time correlation functions and coherent backscattering.

Ge segregation mechanism during Si/Ge multilayer growth

The Ge segregation mechanism during Si/Ge multilayer growth has been investigated using medium-energy ion scattering (MEIS) and computer simulation with the stochastic Monte Carlo technique. We found that Ge atoms segregated not only in the topmost layer but also in the second and third layers. The segregation probability is found to be quadratic in the Ge concentration at the surface. The exchange between the second and the third layer also occurs readily during growth, even although these layers are buried and therefore are not expected to reduce the surface free energy. We found that the activation energy for the site exchange process at the subsurface is lower than the previously believed value, and flip-flop exchange at the subsurface plays a more important role in the Ge segregation process.

Kinetic simulation of molecular beam epitaxial growth dynamics

We kinetically simulate MBE growth using the stochastic Monte Carlo technique within a model that includes surface diffusion of activated Arrhenius type and allows for bulk vacancies and overhangs. We find that the MBE growth exponents are functions of both the diffusion rate (i.e. the temperature) and the deposition rate, and, therefore, there are no unique dynamic exponents universally characterizing the low-temperature kinetically rough growth regime. One can, in principle, obtain layer-by-layer growth at low temperatures by suitably lowering the deposition rate. We present large-scale simulation results on the dynamical coverage, the surface roughness, the RHEED intensity and the defect density to elucidate the kinetic basis of MBE growth within our generalized SOS model.

Numerical studies of epitaxial kinetics: What can computer simulation tell us about nonequilibrium crystal growth?

Computer‐simulation‐based numerical study of nonequilibrium crystal growth is reviewed with the specific emphasis on molecular‐beam epitaxy (MBE). We discuss in detail how computer simulation naturally allows for a quantitative analysis of the effects of various kinetic processes (e.g., surface and bulk diffusion, evaporation, chemical reaction, etc.) on the epitaxial growth quality. Some general features of two extensively employed simulation methods, namely, the molecular dynamics and the stochastic Monte Carlo techniques, are pointed out. We also contrast and compare these techniques, discussing their relative merits and limitations. Predictive abilities of various simulation techniques with respect to MBE growth modes and mechanisms are critically examined. It is shown that general trends as functions of growth parameters such as temperature, substrate orientation, deposition rate, etc. can be qualitatively well understood on the basis of current simulations, however, detailed quantitative predictions...

Surface kinetics driven annealing and phase segregation in non-equilibrium crystal growth

Abstract We simulate the epitaxial growth and annealing of Ge x Si 1− x alloys on a Si(001) substrate using a stochastic Monte Carlo technique. Initiation of phase segregation into Si(001) and Ge(001) layers is obtained in our simulation. This results from the kinetically more active nature of Ge atoms as compared to Si atoms, together with an effect arising from the compressive strain in the alloy layers. We discuss our results in the light of the existing experimental data.

Dynamics of strained-layer epitaxy: Simulation of growth and annealing of GexSi1−x/Si systems

We simulate the epitaxial growth and annealing of GexSi1−x alloys on a Si(001) substrate using a stochastic Monte Carlo technique. Our simulation indicates that: (i) the overgrowth is defect-free for x≤0.2, whereas defect formation is inevitable for x≥0.5; and (ii) initiation of phase segregation into Si and Ge(001) layers is obtained by means of solely surface diffusion effects during growth and annealing. These result from the kinetically more active nature of Ge atoms as compared to Si atoms, together with an effect arising from the compressive strain in the alloy layers.