Thermal etching models

From a knowledge of the absorption coefficient aeff and the threshold fluence Fth, the value of the photochemical contribution to the etch rate, Fph0to> can calculated and subtracted from the measured hole depth for any given fluence. The difference, which according to this model is the thermal etch rate Fthermai will be modeled by Eq. (5) by rewriting it as... 

According to a second class of model, thermal etching is driven by a need to reduce total surface free energy. According to this theory, faceting will take place even in the absence of any net weight loss. The first to suggest a model of... 

Experiments conducted by the same group for the thermal etching of silver in an oxygen atmosphere suggest that evaporation of metal plays little role in the thermal etching process. The thermodynamic model appears to best explain the observations. That is, identical faceted surfaces formed both in the case of suppressed and free evaporation. 

Extensive theoretical work in support of both models of thermal etching was produced before 1970. The theoretical basis for the thermodynamic model was the concept of reduction in total surface energy by the preferential formation of low-energy, low-index planes. The true equilibrium shape of a crystal is the shape with the lowest surface energy, as noted by Curie (36) and Gibbs (37). The thermodynamic models provide no information regarding the process of surface rearrangement. 

There was also a great deal of modeling of transport processes in support of the kinetic model of thermal etching. The basis of these models is that differences in chemical potential lead to mass transport via a number of mechanisms. It is important to note that these models treat the surface as a continuum and do not involve atomic-level mechanisms. 

Powder morphology was investigated using a transmission electron microscope (TEM, Model JEM-IOOCXII). Crystallite size of the powders and grain size of Nd YAG ceramics calcined at different temperatures were calculaied by X-ray diffraction (XRD, model D/maxrA, using nickel-filtered Cu-Ka radiation) patterns from the Scherrer s equation. Microstructures of the fractured and the thermal etched mirror-polished surfaces of Nd YAG specimens were observed by scanning electron microscopy (SEM, Model S-4800). Densities of the samples were measured by the Archimedes draining method. 

Srinivasan etal.,64 in a phenomenological development, split the etch rate into thermal and photochemical components and used zeroth-order kinetics to calculate the thermal contribution to the etch rate. An averaged time-independent temperature that is proportional to the incident fluence was used to determine the kinetic rate constant. The photochemical component of the etch rate was modeled using, as previously discussed, a Beer s law relationship. The etch depth per pulse is expressed, according to this model, in the form... 

Furzikov79 proposed a thermal model to describe the etching rate that led to an inverse square root dependence of the threshold fluence on a modified absorption coefficient, aeff, which includes possible changes in the singlephoton absorption coefficient owing to thermal diffusion. This inverse square root relation is given by... 

Catalytic etching, 41 359, 383-384 definition, 41 360-361 in low earth orbit, 41 414—415 models, 41 359, 360-362 plasma etching, 41 407-414 thermally generated free radicals, 41 406-407... 

This model is quite universal providing that the ion-induced reaction rates prevail on the thermal reaction rates. It has been thoroughly discussed for Si and SiCF etching by means of beam experiments [75-77], and has been checked in plasma environment [78]. It is also verified for other systems Si in Cl2, [79], SiGe alloys in SF5 [80], or InP in CH4—H2 [81]. 

These apparent contradictions can be rationalized in terms of a model which incorporates plasma-induced polymerization along with depolymerization. PBS has long been known to exhibit a marked temperature-dependent etch rate in a variety of plasmas. This is clearly seen in the previously published Arrhenius plots (3,7) for two different plasma conditions (Figure 1). This dependence is characteristic of an etch rate that is dominated by an activated material loss as would occur with polymer depolymerization. The latter also greatly accelerates the rate of material loss from the film. Bowmer et al. (10-13) have shown in fact that poly(butene-l sulfone) is thermally unstable and degrades by a depolymerization pathway. A similar mechanism had been proposed by Bowden and Thompson (1) to explain dry-development (also called vapor-development) under electron-beam irradiation. 

Since the electron transfer from the conduction band into the surface state (Eq. 67) can be very fast and the corresponding rate may be determined by the thermal velocity of electrons toward the surface, it has to be assumed that the initial chemical etching reaction (66) is even faster. However, it is not clear whether this assumption is correct. Very recently it has been found that also the reduction of protons (H2-formation) at n-GaAs is a very fast reaction. The current potential dependence can actually be described by the thermionic emission model (see Eq. (65)) [142]. This result indicates that the electron transfer can occur at much higher rates if the electron acceptor is adsorbed on the surface. This assumption is supported by recent results reported by Nozik [143]. He repeated his fluorescence decay measurements by using nitrobenzene as an electron acceptor and found a much lower rate than for ferrocence. Nozik assumed that the high rate constant for ferrocence may also be due to adsorption. 

Samalam [43] modeled the convective heat transfer in water flowing through microchannels etched in the back of silicon wafers. The problem was reduced to a quasi-two dimensional non-linear differential equation under certain reasonably simplified and physically justifiable conditions, and was solved exactly. The optimum channel dimensions (width and spacing) were obtained analytically for a low thermal resistance. The calculations show that optimizing the channel dimensions for low aspect ratio channels is much more important than for large aspect ratios. However, a crucial approximation that the fluid thermophysical properties are independent of temperature was made, which could be a source of considerable error, especially in microchannels with heat transfer. 

In a slightly different angle, Kendall reasoned that beeause the 111 surfaee is oxidized thermally more rapidly than other low-index surfaees, the silicon surface ean be covered with a silicon oxide (or a hydrated silieon oxide) during etehing in aqueous solution, mueh faster than other planes. The formation of the oxide film passivates the (111) plane and blocks the dissolution reaetions. This model implies that the etch rate of a (111) surface should be similar to that of silieon oxide in KOH solu-... 

The plasma etch rates of the SOG films were determined In an MRC Model 51 RIE parallel plate etcher which had 6 in. diameter electrodes. The electrode spacing was 2 in. and the substrates were placed on the powered electrode which was water cooled. The chamber pressure was about 200 mTorr for plasma etching and 10-15 mTorr for reaction ion etching (RIE). A net total power of 50W was used in each experiment. In contrast to their wet etch behavior, the SOG films etch only slightly faster than thermal Si02 in CF4 + O2 plasmas. This Is so since plasma etch rates are determined to a greater... 

Let us consider silver particles prepared on a model system for our experimental studies. The clusters were prepared inside the spectrometer by evaporation from thermally heated tungsten sources using high-purity wires. The substrate used was carbon prepared by prior evaporation onto mica with the technique and equipment described by Hamilton et al. (4 ). Upon introduction into the spectrometer, the carbon film was ion-etched with Xe, to remove traces of or S. These samples remained clean for a few days before use as substrates for the evaporated metal. 

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