Reply to Comment on "Exact analytic solution for the generalized Lyapunov exponent of the 2-dimensional Anderson localization"

We reply to comments by P.Marko$\breve{s}$, L.Schweitzer and M.Weyrauch [preceding paper] on our recent paper [J. Phys.: Condens. Matter 63, 13777 (2002)]. We demonstrate that our quite different viewpoints stem for the different physical assumptions made prior to the choice of the mathematical formalism. The authors of the Comment expect \emph{a priori} to see a single thermodynamic phase while our approach is capable of detecting co-existence of distinct pure phases. The limitations of the transfer matrix techniques for the multi-dimensional Anderson localization problem are discussed.Comment: 4 pages, accepted for publication in J.Phys.: Condens. Mat

Annihilation of Immobile Reactants on the Bethe Lattice

Two-particle annihilation reaction, A+A -> inert, for immobile reactants on the Bethe lattice is solved exactly for the initially random distribution. The process reaches an absorbing state in which no nearest-neighbor reactants are left. The approach of the concentration to the limiting value is exponential. The solution reproduces the known one-dimensional result which is further extended to the reaction A+B -> inert.Comment: 12 pp, TeX (plain

Analytical realization of finite-size scaling for Anderson localization. Does the band of critical states exist for d>2?

An analytical realization is suggested for the finite-size scaling algorithm based on the consideration of auxiliary quasi-1D systems. Comparison of the obtained analytical results with the results of numerical calculations indicates that the Anderson transition point is splitted into the band of critical states. This conclusion is supported by direct numerical evidence (Edwards and Thouless, 1972; Last and Thouless, 1974; Schreiber, 1985; 1990). The possibility of restoring the conventional picture still exists but requires a radical reinterpretetion of the raw numerical data.Comment: PDF, 11 page

The Reaction-Diffusion Front for A+B→∅A+B \to\emptyset in One Dimension

We study theoretically and numerically the steady state diffusion controlled reaction $A+B\rightarrow\emptyset$, where currents $J$ of $A$ and $B$ particles are applied at opposite boundaries. For a reaction rate $\lambda$, and equal diffusion constants $D$, we find that when $\lambda J^{-1/2} D^{-1/2}\ll 1$ the reaction front is well described by mean field theory. However, for $\lambda J^{-1/2} D^{-1/2}\gg 1$, the front acquires a Gaussian profile - a result of noise induced wandering of the reaction front center. We make a theoretical prediction for this profile which is in good agreement with simulation. Finally, we investigate the intrinsic (non-wandering) front width and find results consistent with scaling and field theoretic predictions.Comment: 11 pages, revtex, 4 separate PostScript figure

Segregation in diffusion-limited multispecies pair annihilation

The kinetics of the q species pair annihilation reaction (A_i + A_j -> 0 for 1 <= i < j <= q) in d dimensions is studied by means of analytical considerations and Monte Carlo simulations. In the long-time regime the total particle density decays as rho(t) ~ t^{- alpha}. For d = 1 the system segregates into single species domains, yielding a different value of alpha for each q; for a simplified version of the model in one dimension we derive alpha(q) = (q-1) / (2q). Within mean-field theory, applicable in d >= 2, segregation occurs only for q < 1 + (4/d). The only physical realisation of this scenario is the two-species process (q = 2) in d = 2 and d = 3, governed by an extra local conservation law. For d >= 2 and q >= 1 + (4/d) the system remains disordered and its density is shown to decay universally with the mean-field power law (alpha = 1) that also characterises the single-species annihilation process A + A -> 0.Comment: 35 pages (IOP style files included), 10 figures included (as eps files

Symmetry and species segregation in diffusion-limited pair annihilation

We consider a system of q diffusing particle species A_1,A_2,...,A_q that are all equivalent under a symmetry operation. Pairs of particles may annihilate according to A_i + A_j -> 0 with reaction rates k_{ij} that respect the symmetry, and without self-annihilation (k_{ii} = 0). In spatial dimensions d > 2 mean-field theory predicts that the total particle density decays as n(t) ~ 1/t, provided the system remains spatially uniform. We determine the conditions on the matrix k under which there exists a critical segregation dimension d_{seg} below which this uniformity condition is violated; the symmetry between the species is then locally broken. We argue that in those cases the density decay slows down to n(t) ~ t^{-d/d_{seg}} for 2 < d < d_{seg}. We show that when d_{seg} exists, its value can be expressed in terms of the ratio of the smallest to the largest eigenvalue of k. The existence of a conservation law (as in the special two-species annihilation A + B -> 0), although sufficient for segregation, is shown not to be a necessary condition for this phenomenon to occur. We work out specific examples and present Monte Carlo simulations compatible with our analytical results.Comment: latex, 19 pages, 3 eps figures include

Основные подходы к моделированию формирования фоторезистивной маски в вычислительной литографии

The article gives an overview of the main currently used models for the formation of photoresist masks and the problems in which they are applied. The main stages of «full physical» modeling of mask formation are briefly considered in the case of both traditional DNQ photoresists and CA photoresists. The concept of compact models (VT5 and CM1), which predict the contour of the resist mask for a full-sized device topology is considered. Examples of some calculations using both full physical modeling and compact models are given. Using a full physical modeling of the resist mask formation the lithographic stack was optimized for a promising technological process. The optimum thickness ratios for the binary BARC used in the water immersion lithographic process are found. The problem of determining the optimal number of calibration structures that maximally cover the space of aerial image parameters was solved. To solve this problem, cluster analysis was used. Clustering was carried out using the k-means method. The optimal sample size was from 300 to 350 structures, the mean square error in this case is 1.4 nm, which slightly exceeds the noise of the process for 100 nm structures. Using SEM images for calibrating the VT5 model allows reducing the standard error of 40 structures to 1.18 nm.В статье дан обзор основных моделей формирования фоторезистивной маски, используемых в настоящее время, и задач, в которых они применяются. Кратко рассмотрены этапы «полного» моделирования формирования маски, основанного на физико-химических принципах, в случае как традиционных нафтохинондиазидовых фоторезистов, так и фоторезистов с химическим усилением. Рассмотрена концепция основных применяемых в настоящее время компактных моделей, предсказывающих контур фоторезистивной маски для полноразмерной топологии изделия, а именно, моделей VT5 (Variable Threshold 5) и CM1 (Compact Model 1). Приводятся примеры некоторых расчетов с использованием как полного моделирования формирования маски, так и компактных моделей. При помощи полного моделирования формирования фоторезистивной маски был оптимизирован литографический стек для перспективного технологического процесса. Найдены оптимальные соотношения толщин для бинарного антиотражающего слоя, применяемого в литографическом процессе с водной иммерсией. При калибровке компактной модели VT5 решена задача определения оптимальной выборки калибровочных структур, максимально охватывающих пространство параметров оптического изображения, используя при этом минимальное количество структур. Для решения указанной задачи использовался кластерный анализ. Кластеризация проводилась методом k-средних. Оптимальный размер выборки составил от 300 до 350 структур, среднеквадратичная ошибка при этом составляет 1,4 нм, что незначительно превышает шум технологического процесса для 100 нм структур. Использование СЭМ-контуров при калибровке модели VT5 позволяет снизить среднеквадратическую ошибку по 40 структурам до 1,18 нм

Fluctuation Kinetics in a Multispecies Reaction-Diffusion System

We study fluctuation effects in a two species reaction-diffusion system, with three competing reactions $A+A\rightarrow\emptyset$, $B+B\rightarrow\emptyset$, and $A+B\rightarrow\emptyset$. Asymptotic density decay rates are calculated for $d\leq 2$ using two separate methods - the Smoluchowski approximation, and also field theoretic/renormalisation group (RG) techniques. Both approaches predict power law decays, with exponents which asymptotically depend only on the ratio of diffusion constants, and not on the reaction rates. Furthermore, we find that, for $d<2$, the Smoluchowski approximation and the RG improved tree level give identical exponents. However, whereas the Smoluchowski approach cannot easily be improved, we show that the RG provides a systematic method for incorporating additional fluctuation effects. We demonstrate this advantage by evaluating one loop corrections for the exponents in $d<2$, and find good agreement with simulations and exact results.Comment: LaTeX file (41 pages) + 13 postscript figures, uuencode