ТЕОРЕТИЧЕСКИЕ ОСНОВЫ ХИМИЧЕСКОЙ ТЕХНОЛОГИИ, 2013, том 47, № 5, с. 551-557
УДК 519.245:621
MONTE CARLO MODELLING OF WET-CHEMICAL LITHOGRAPHY WITH MASKS
© 2013 г. A. P. Reverberi, B. Fabiano*, V. G. Dovi, V. P. Meshalkin**
Department of Chemistry and Industrial Chemistry, University of Genova, via Dodecaneso 31, 16145 Genova, Italy *Department of Civil, Chemical and Environmental Engineering, University of Genova, via Opera Pia 15, 16145 Genova, Italy
**Mendeleev University of Chemical Technology of Russia, Moscow vpmeshalkin@gmail.com Received 13.03.2013
We propose Monte Carlo simulation of the etching process in two dimensions for the manufacture of microchannels and microcavities on a solid substrate. The method combines the effect of two different regimes based on diffusion-limited disaggregation and reaction-limited erosion, respectively. Besides, the role of the selectivity in site extraction is taken into account to foresee the effects of the temperature of the eroding bath. This technique proves to be a valid alternative to more complex analytical methods to describe surface decay processes in the presence of overhangs. The relevant geometries of the etched surfaces are analyzed, and other statistical properties of the cavities are discussed and compared to the ones predicted by continuum models.
DOI: 10.7868/S0040357113050138
INTRODUCTION
In recent years, the manufacture of micro heat exchangers and microreactors assumed an increasing importance owing to their applications in the productions of fine chemicals, in microelectronics and robotics [1—3]. Generally, the aforementioned apparatuses require a particularly accurate realization of assembly details which can be mechanically realized with great difficulty. Namely, the lower is the scale of geometrical details required in microreactors, the higher are the costs of production by mechanical processes, with a dramatically nonlinear correlation. This fact represents a serious constraint that limits the applicability of physical ablation techniques, such as plasma etching and ion-beam lithography.
Wet-chemical processes of solid ablation represent a valid and cheap alternative to the physical techniques in many fields of micromechanical technologies, and their use is getting almost irreplaceable in nanotech-nology [4]. By the way, nanoparticle synthesis by chemical methods is getting more and more important in bottom-up processes, while top-down techniques, which are strictly related to wet-chemical etching, still represent a challenging and promising aspect of research [5].
Chemical disaggregation refers generally to two main aspects of surface depletion defined as isotropic and anisotropic etching. The first one describes a process of disaggregation where the extraction of an atom from its surface is independent of its chrystallographic orientation. This process typically occurs when the solid is amorphous or it has a randomly oriented structure. On the opposite, anisotropic etching characteriz-
es a solid substrate with intrinsically ordered inner structure, often required in semiconductor technology [6].
The modelling of both the aforementioned processes relies upon a continuum or a discrete approach. The first one consists of a description of the solid surface whose geometrical properties, such as the height of the corrugated surface, are the dependent variables of a deterministic or stochastic differential model.
Kuiken [7] was the first who cast in a rigorous formalism the continuum modelling of a deterministic disaggregation process in one dimension, and an asymptotic solution was proposed in case of diffusion-limited disaggregation. Shin and Economou [8, 9] modelled surface etching in different cases of diffusion and reaction-controlled regimes in two dimensions where the etchant concentration c is described by the classical Fick's second law equation and the boundary conditions at the moving surface of the eroded solid are cast in the following form:
Vc • n = -R(c)l D, (1)
where n is the unit vector normal to the surface, D is the diffusion coefficient of the etchant in the solution and R(c) is the intrinsic kinetic term of solid disaggregation. In case of linear kinetics, R(c) = Kc and the ratio K/D is the tuning parameter that triggers the diffusion- or reaction-limited process control, where K is the chemical kinetic constant.
The position of the moving eroded front is determined together with the resolution of the concentration field and the local speed of the moving boundary
depends on the concentration gradients at the surface as follows:
MD dc.
MD dc
(2)
P, dx p, dy
where M and are the molecular weight and the density of the solid, respectively.
In the total concentration approach of Rath and co-workers [10], the position of the etch front is captured without using a moving boundary technique. Other approaches in terms of mathematical methods have been exhaustively summarized by the same author [11].
While a deterministic approach in continuum models allows to determine macroscopic aspects of the etch front, such as the under etching length under a mask and the lowest position of the eroded profile, a stochastic approach in continuum models allows to foresee microscopic properties of the etch front, such as the rugosity and its scale laws.
In a stochastic differential scheme applied to growth and erosion processes, the time trend of the etch front height h depends on a group of space differential operators as
dh = ¥ (V2h,(Vh)2, V 4h,...) + n(x, t), (3)
where n is the space and time uncorrelated Gaussian random noise. It has been shown that a relaxation term in a growth process, such as V2h, is related to selective disaggregation in etching processes [12]. Despite this approach is very promising for its capability of capturing the atomistic phenomena that condition local geometries particularly important in semiconductor etching, the inner computational burden for the solution of Eq. (3) has actually limited its extended application in surface disaggregation [13]. In fact, as a rule of thumb for continuum models in physical chemistry, the search for an analytical or numerical solution of deterministic schemes is generally preferred to the stochastic differential approach [14—20].
In the field of discrete simulations, Monte Carlo techniques represent a good trade-off between easiness in implementation and reliability of results, and this is probably the reason of their large application in surface disaggregation processes [21]. This approach has been adopted both for anisotropic [6] and isotropic etching of metals and semiconductors. In particular, this technique proved to be useful to study the limiting conductivities of metal strips and wires subject to total depletion up to disconnection. It was shown that the percolation effects in the fluctuation of electronic transport properties of nanocontacts may have a role comparable to the one of quantum fluctuation [22]. Besides, as in the stochastic differential models, Monte Carlo techniques allow one to estimate the effects of local atomistic rules on macroscopic shape and patterns that are basically important in the production of optical and mechanical devices. In this context, the
use of masks as inert layers preventing the chemical attack of the underlying material is a commonly adopted strategy.
In this paper, we present an on-lattice two-dimensional simulation of a chemical attack to a solid surface partially covered by an inert mask. The average surface shape and other macroscopic properties of the eroded front are studied according to the microscopic rules of etcher motion and of site extraction.
The paper is divided as follows. First, we outline the method and we give some details of the rules adopted in this technique. Then, we present the results according to the onset of different disaggregation regimes tuned by three different modulation parameters. Finally, we draw the conclusions and we trace the direction for future works.
THE MODELS
An array of N x N sites represents the embedding space where a site A occupied by the solid substrate at coordinate (i, j) is labelled assigning A(i, j) = 1 while an empty site occupied by the etchant solution is labelled by A(i, j) = 0. The elements with A(i, j) = 2 indicate the sites belonging to an inert mask which is not subject to erosion. The mask covers all the solid surface, except for a zone a < x < b, where the solid is exposed to the chemical attack.
Three tuning parameters, namely, a, P and y, each belonging to a range [0, 1], are fixed at the beginning of the simulation in order to model the etching control regime, the motion of an etchant particle and the site selection criteria of extraction, respectively. To choose which erosion technique will be adopted at the following time step, a random number z £ [0, 1] with uniform distribution is generated and hence:
if z < a, diffusion-limited erosion (DLE) will occur;
if z > a, reaction-limited erosion (RLE) will occur.
DLE scheme. In this case, an etchant molecule represented by a moving walker is launched at a fixed distance above the eroded surface and it walks step by step only on nearest-neighbours empty sites. The initial configuration is sketched in Fig. 1, where two different launching options have been visualized. Suppose that the etcher is on a site A(i, j) surrounded by four nearest-neighbours with A(i + 1, j) = A(i — 1, j) = = A(i, j + 1) = A(i, j — 1) = 0, that is by four empty sites. To decide the next position of the etcher, a random number z £ [0, 1] with uniform distribution is generated and the following decisions are taken:
if 0 < z < (1 - 3P/4), (i, j) ^ (i, j - 1) (downward jump);
if (1 - 3p/4) < z < (1 - P/2), (i, j) ^ (i, j + 1) (upward jump);
if(1 - P/2) < z < (1 - P/4), (i, j) ^ (i + 1, j) (right-ward jump);
if (1 - P/4) < z < 1, (i, j) ^ (i - 1, j) (leftward jump).
(a)
Reflecting barrier
(b)
Etcher □
Fig. 1. Scheme
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