Statistical insights into the reaction of fluorine atoms with silicon

The dependences of silicon etching rate on the concentration of F atoms are investigated theoretically. The nonlinear regression analysis of the experimental data indicates that the reaction of F atoms with silicon is 2nd overall order reaction. The relationship between overall reaction order and kinetic reaction order is established using the etching rate equation. It is found that kinetic reaction order monotonically decreases with the increase in concentration of F atoms due to the increased surface coverage. Surface passivation by the reaction products is not observed under the investigated experimental conditions.


Rimantas Knizikevičius
The dependences of silicon etching rate on the concentration of F atoms are investigated theoretically. The nonlinear regression analysis of the experimental data indicates that the reaction of F atoms with silicon is 2nd overall order reaction. The relationship between overall reaction order and kinetic reaction order is established using the etching rate equation. It is found that kinetic reaction order monotonically decreases with the increase in concentration of F atoms due to the increased surface coverage. Surface passivation by the reaction products is not observed under the investigated experimental conditions.
Scanning tunnelling microscopy (STM) is widely used to analyse chemical reactions taking place on the silicon surface. STM enabled to measure the etching rate dependences on the initial concentration of SiCl, SiBr, and SiI radicals 1,2 . During the experiments, silicon substrates were dosed with molecular halogens at room temperature in order to obtain the desired initial concentration. Subsequently, silicon substrates were heated because chemical reactions between the chemisorbed radicals occur only at elevated temperatures. STM provided useful information about the reaction and desorption pathways 3 and confirmed that silicon dihalides are the final reaction products. The same experimental procedure is used to investigate chemical reaction between SiF radicals on the silicon surface. The etching rate is measured only at low initial concentration of SiF radicals because fluorine atoms due to small atomic radius penetrate into the silicon lattice 4,5 . The experimental measurements 6 shown that at temperature 825 K planar removal of silicon atoms occurs together with multilayer pitting, which result in the increased surface roughness.
Steady-state reaction of F atoms with silicon is investigated using large array of diagnostic techniques 7 . The most common observations are following: (a) SiF 4 molecules detected by mass spectrometry of exhaust species 8 ; (b) SiF x (x ≤ 3) radicals found on the etched Si surface using X-ray photoemission spectroscopy 9 ; (c) SiF 2 molecules detected by chemiluminescence 10 and laser-induced fluorescence spectroscopy 11 ; (d) SiF 2 molecules polymerize on the surface 12 . (e) SiF radicals passivate the Si surface 13 ; The experimental measurements are subsequently analysed using theoretical models. Simplest theoretical models successfully describe specific phenomena 14 , while simulators provide more complete view of the etching process.
The overall reaction order is determined using isothermal dependences of silicon etching rate on the concentration of F atoms. The mean lifetime of F atoms under most common plasma processing conditions is several microseconds 15 . Within that time F atoms must diffuse from plasma to the silicon substrate and chemisorb on the surface. The flow of F atoms always has other species admixed. This affects surface coverage and changes reactivity of silicon substrates 16 . The number of artefacts is significantly reduced by using F 2 plasma as the source of fluorine atoms. In this case, special design of the experimental setup is required to handle highly reactive F atoms and F 2 gas. Only two research groups successfully performed the experiment: 1. in the work 17 , fluorine atoms were produced by the dissociation of 98.5% pure F 2 gas using 14 MHz discharge in an alumina tube 50 cm upstream of the reaction cell.  18 , fluorine atoms were produced by the dissociation of 99% pure F 2 gas using 2.45 GHz discharge in an alumina tube outside the reaction chamber. The flow rate of F 2 gas was kept constant at 8 sccm, while pumping speed was adjusted by the throttle valve. Concentration of F atoms varied from 1.1 × 10 12 cm −3 at 150 W to 3.6 × 10 14 cm −3 at 220 W discharge power. The concentration of F atoms was measured by gas-phase titration with H 2 gas. Si(100) substrates were placed in the stainless steel reaction chamber. The temperature of silicon substrates was kept constant at 300 K. The etching rate was calculated by dividing the etched trench depth by the etching time.
The experimental scientists described steady-state etching rate using the empirical equation V = ε † [F] γ , where ε † is the rate constant and γ is the kinetic reaction order. In the proposed equation, kinetic reaction order describes how silicon etching rate depends on the concentration of F atoms. US-based research group 17 assumed that the measured dependence is linear, and the kinetic reaction order is equal to 1. Japan-based research group 18 used plot log 10 V versus log 10 [F] to determine the kinetic reaction order. The linear plot provided the same averaged value of kinetic reaction order. However, the method used to determine kinetic reaction order is unreliable in the presence of intermediate reaction products. It is important to note that both research groups wrongly assumed that formed reaction products immediately desorb. As a result, the etching process was described using single activation energy, derived from the rate constant ε † . At that time, the elementary processes were well documented 19 but their activation energies in most cases were unknown. This resulted in the widespread assumption that the etching process requires insignificant activation energy. Recent theoretical calculations 20,21 indicate that desorption process must be included in the description of the experimental data because of the high energetic barrier.
In this work, the isothermal dependences of silicon etching rate on the concentration of F atoms are reanalysed using the chemical kinetics. The overall reaction order is determined using nonlinear regression analysis of the experimental data. The etching rate equation enabled to establish relationship between overall reaction order and kinetic reaction order. It is found that kinetic reaction order equals to the Si surface fraction not covered with adsorbate.

Theory
The enthalpy changes during the reactions of F atoms with silicon on the surface and silicon fluorides in the gas phase are presented in Table 1. In order to derive the overall reaction order, the following reactions of F atoms with silicon substrate are considered: where 1 ≤ n ≤ 4 is the partial reaction order for F atoms. The overall reaction order is equal to m = n + 1 . The reaction rate is described using transition-state theory by the reaction rate constant k r . The mean reaction time is equal to τ r = k −1 r p −n F , where p F is the partial pressure of F atoms. The etching process takes place when the reaction products desorb. The desorption rate constant according to the transition-state theory is equal to where h is the Planck constant, k is the Boltzmann constant, T is the temperature, and E d is the desorption activation energy. The mean desorption time is equal to τ d = ω −1 . The relative error of desorption rate constant is equal to When the substrate temperature is determined accurately, the absolute error of desorption activation energy reaches maximum value (1) Si(s) + nF g → SiF n (a), Table 1. The reaction enthalpies, measured in electronvolts, at standard conditions.

SiF(g) SiF 2 (g) SiF 3 (g) SiF 4 (g)
Si ( www.nature.com/scientificreports/ Otherwise, when the desorption activation energy of the reaction product is determined accurately, the absolute error of temperature reaches maximum value SiF n species are included in the adsorbed layer of one-monolayer thickness. The relative concentration of SiF n species in the adsorbed layer is equal to c = [SiF n ] C , where C = 6.78 × 10 18 m −2 is the planar density of Si (100) substrates. The following differential equation includes rate expressions of the processes mentioned earlier and describes the concentration kinetics in the adsorbed layer: where β = 1 − is the surface fraction not covered with adsorbate and = c is the surface coverage. The steady-state concentration of SiF n species is equal to The steady-state etching rate, which is equal to the desorption rate of formed SiF n species, is calculated using the following equation: Kinetic reaction order for the above mentioned chemical reactions under steady-state conditions is calculated using the following equation: It is important to note that kinetic reaction order monotonically decreases with the increase in partial pressure of F atoms. Meanwhile, the overall reaction order does not depend on the partial pressure of F atoms.

Results and discussion
The reaction of fluorine atoms with silicon at constant temperature is investigated using the nonlinear regression of the experimental data. The experimental and theoretical dependences of silicon etching rate on the partial pressure of F atoms are shown in Fig. 1. In both cases, best quality fits are obtained using first partial reaction order for F atoms, see Table 2. Moreover, the values of R-square and adjusted R-square indicate that the theoretical model better fits the experimental data 18 . The goodness-of-fit parameters are thoroughly described in the work 22 . According to the chemical kinetics, the reaction of F atoms with silicon is 2nd overall order reaction. This means that SiF radicals are the final reaction product. However, SiF radicals tend to passivate the Si surface, and suppress the etching rate at high partial pressure of F atoms in the plasma 13 . The experiment was performed in the parallel-plate plasma deposition system because of high working pressure (up to 300 Pa). While, the theoretical results showed that the etching rate is equal to the desorption rate of formed SiF 2 molecules. Let us adopt the theoretical results presented in the work 13 and assume that SiF 2 molecules prevail in the flux of desorbing c st = k r p n F k r p n F + ω . where SV is the surface vacancy and DB is the dangling bond. The conversion of SiF radicals into SiF 2 molecules in the presence of surface defects is confirmed experimentally 6 . Sophisticated experimental equipment is required in order to detect desorbing SiF 2 molecules because they are stable only at elevated temperatures. At room temperature, SiF 2 molecules, regardless of their relatively high mean lifetime in the gas phase 27 , are converted into SiF 4 molecules. Therefore, simplest measurement methods such as mass spectrometry struggle to detect SiF 2 molecules. The temperature-programmed desorption experiments indicate that mean lifetime of SiF 3 radicals in the gas phase is much lower than that of SiF 2 molecules 28 .
The nonlinear regression analysis of the experimental data yields kinetic parameters associated with Si + nF → SiF n reactions. The reaction and desorption rate constants, determined using different partial reaction orders for F atoms, are presented in Table 3. The reaction rate constants describe how silicon etching rate increases with the increase in partial pressure of F atoms. The reaction rate constants are determined accurately using first partial reaction order. However, the relative error of the reaction rate constants sharply increases with the increase in partial reaction order. The reaction activation energies are not calculated because high reaction activation entropy can lead to significant errors and misinterpretations. Desorption rate constants describe asymptotical approach of the etching rate to the saturation regime. The desorption rate constants are determined accurately except one at temperature 296 K using first partial reaction order for F atoms. In this particular case, the predicted dependence is nearly linear in the considered range of partial pressure of F atoms, and the fitting program yields large relative error of the desorption rate constant. The average desorption activation energy of the reaction product determined using first partial reaction order is equal to (0.640 ± 0.007) eV.
In work 29 , desorption activation energy of SiF 4 molecules at temperature 296 K was evaluated using fourth partial reaction order for F atoms. It was determined that the desorption activation energy of SiF 4 molecules is equal to (0.66 ± 0.03) eV . The obtained theoretical results showed that desorption of the reaction product requires significant activation energy. However, the reaction product prevailing in the flux of desorbing species was not identified correctly. In work 20 , desorption activation energy of SiF 2 molecules at temperature 376 K was evaluated using first partial reaction order for F 2 molecules. It was determined that desorption activation energy   www.nature.com/scientificreports/ of SiF 2 molecules is equal to (0.815 ± 0.010) eV . It is important to note that the difference between determined desorption activation energies is equal to 0.155 eV. This corresponds to the difference in the substrate temperature 1,800 K, which exceeds the silicon melting point at standard conditions. However, melting of the silicon substrates was not observed experimentally. The experimental data 19 indicate that activation energies of elementary processes increase with the increase in temperature during chemical etching of silicon in F 2 environment. At room temperature, diffusion of Si atoms on the surface is strongly anisotropic. Diffusion activation energy for Si atoms along dimer rows is much lower than that between dimer rows 30,31 . The reconstruction of silicon surface becomes pronounced at temperatures above 350 K. This phenomenon reduces diffusion of Si atoms along the dimer rows below the expected thermal values 32 , and increases the activation energies of elementary processes. Let us consider the etching process using another statistical approach. The reaction constant shows how many Si atoms are removed from the surface by single F atom where �(SiF 2 ) = ω[SiF 2 ] is the flux of desorbing SiF 2 molecules, �(F) = p F (2πm F kT) −1 / 2 is the flux of F atoms to the silicon surface, and m F is the mass of F atom. It is important to note that reaction constant depends on the partial pressure of F atoms. At extremely low pressure, the reaction constant reaches maximum value According to Eq. (6), the ratio ε ε 0 is equal to the surface fraction not covered with adsorbate The dependences of ratio ε ε 0 on the partial pressure of F atoms are shown in Fig. 2. The maximum value of reaction constant during the experiment 17 is equal to ε 0 = 2.50 × 10 −4 (99.95% of F atoms incident to the atomically clean silicon surface are reflected). Meanwhile, the maximum value of reaction constant during the experiment 18 is equal to ε 0 = 1.72 × 10 −2 (96.56% of F atoms incident to the atomically clean silicon surface are reflected). Although the experimental conditions are very similar, the reaction constants ε 0 differ 69 times. The difference can be caused by measurement errors made during the gas phase titration and by exposure of silicon substrates to the plasma radiation with the subsequent increase in temperature 33,34 . The substrate temperature can also increase due to poor thermal contact between the substrate and the substrate holder 35 . Highest absolute error of temperature (about 6 K) is obtained during nonlinear regression of the experimental data 17 . However, the temperature uncertainty cannot explain the observed difference between the reaction constants.
Kinetic reaction order is calculated using first partial reaction order for F atoms. SiF radicals are formed rapidly in the adsorbed layer but subsequently tend to passivate the silicon surface. The surface passivation is described by negative values of the kinetic reaction order. According to the model, kinetic reaction order monotonically decreases with the increase in partial pressure of F atoms but negative values of the kinetic reaction order are not achieved. The theoretical dependences of kinetic reaction order on the partial pressure of F atoms, presented in Fig. 2, indicate that concentration of SiF radicals in the adsorbed layer is insufficient to passivate the surface. SiF radicals are converted into SiF 2 molecules, which subsequently desorb. In the case of first partial reaction order for F atoms, the same equation describes: (a) the dependence of kinetic reaction order on the partial pressure of F atoms; (b) the dependence of ratio ε ε 0 on the partial pressure of F atoms; The mean reaction time and the mean desorption time provide an insight into the etching-rate limiting process. According to the model, mean reaction time reciprocally decreases with the increase in partial pressure of F atoms, while mean desorption time does not depend on the partial pressure of F atoms. The theoretical results, presented in Fig. 3, indicate that F atoms are much more reactive towards the silicon substrates during the experiment 18 . The difference can be caused by measurement errors made during the gas phase titration and by exposure of silicon substrates to the plasma radiation. However, higher mean desorption time of SiF 2 molecules in the experiment 18 indicates lower temperature of the silicon substrates (by about 3 K). This means that the exposure of silicon surface to the plasma radiation during the experiment 18 can be neglected. The difference between the experiments 17,18 is caused by the measurement errors made during the gas phase titration of F atoms. The statement is supported by the different ranges of partial pressure of F atoms used in the experiments.
The experimental results 17,18 converge into single curve at the same desorption rate constant. Kinetic reaction order, ratio ε ε 0 , and surface fraction not covered with adsorbate are plotted versus reciprocal of the mean reaction time in Fig. 4. The data points shown on the curve indicate theoretically calculated values of three different parameters at experimentally measured partial pressure of F atoms. It is important to note that the last two data points corresponding to the kinetic reaction order of about 0.4 were intentionally omitted from the plot log 10 V versus log 10 [F] in order to obtain the linear dependence by the Japan-based research group 18 . All other data points show that kinetic reaction order varies from 1 to approximately 0.7. This indicates that up to 30% of the silicon surface is covered by adsorbate. The coverage has little influence on the experimentally measured etching rate. Therefore, the experimental scientists tend to use term "pseudo-first kinetic reaction order". In most cases, the term is shortened to the "pseudo-first reaction order" 25,36 , although this not correct from the view point of chemical kinetics.  www.nature.com/scientificreports/ Finally, the subtle difference between the proposed model and Langmuir adsorption model must be considered. Langmuir isotherms are obtained using the following assumptions 37 : (a) surface with the adsorption sites is atomically flat; (b) adsorption sites have the same adsorption activation energy; (c) species adsorb only into an immobile state; (d) surface coverage is one-monolayer thickness; (e) adsorbed species are chemically inert.
The most notable difference between the proposed model and Langmuir adsorption model is that adsorbate diffuses on the surface. When single plasma component adsorbs on the surface, both models provide identical equations. The difference induced by the adsorbate diffusion becomes pronounced when two or more plasma components adsorb on the surface 38 . Additionally, the reaction between SiF radicals in the adsorbed layer was deduced by comparison of the obtained theoretical results with the experimental measurements.

Conclusions
Chemisorption of F atoms on the silicon surface does not occur instantly. At room temperature, at least 96% of F atoms incident to the atomically clean silicon surface are reflected. The nonlinear regression analysis of the experimental data indicates that the reaction of F atoms with silicon is 2nd overall order reaction. The volatile reaction product is formed during the reaction 2SiF(a) SV/DB −→ SiF 2 (a) + Si(s) in the adsorbed layer. Kinetic reaction order monotonically decreases with the increase in partial pressure of F atoms due to the increased surface coverage. Surface passivation by SiF radicals is not observed under the investigated experimental conditions.