Silicon Etching during the HFCVD Diamond Growth - American

The silicon etching that occurs during the CVD diamond growth has been investigated as a ... depth, and angular distributions of the etch pits were re...
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J. Phys. Chem. B 1998, 102, 4856-4864

ARTICLES Silicon Etching during the HFCVD Diamond Growth J. C. Arnault,* S. Hubert, and F. Le Normand Groupe Surfaces-Interfaces, Institut de Physique et Chimie des Mate´ riaux de Strasbourg, IPCMS-GSI, UMR 7504, Bat 69, 23, rue du Loess, 67037 Strasbourg, France ReceiVed: February 2, 1998

The silicon etching that occurs during the CVD diamond growth has been investigated as a function of the methane content in the gas phase by SEM and AFM in the tapping mode on pristine Si(111) surfaces. Size, depth, and angular distributions of the etch pits were recorded. We evidence the strong effect of the carbon content on the etching process. The silicon etching is slightly increased with addition of 0.1-0.25% of methane in the feed gas, and depletes with larger addition of carbon. This etching occurs easily along Si(100) directions. This is explained by the better stability on (100) planes of the precursor SiH2 to remove silane than on (111) planes. A modelization of the process points out the balance between the drop of the atomic hydrogen concentration in the gas phase and the inhibition of hydrogen bulk diffusion into silicon when increasing the methane content and covering the surface with carbon. This explains the occurrence of a maximum of silicon etching. Possible consequences on the diamond nucleation process are then put forward. It is thus expected that the etching of the silicon is generally detrimental as generating low-density silicon surface such as Si(100) where diamond nucleation is inhibited.

Introduction The interaction of atomic hydrogen with clean silicon surfaces has been recently extensively studied.1-3 It is now welletablished that etching of a silicon surface by atomic hydrogen proceeds by the breaking of Si-Si bonds. This leads to the formation of higher surface hydrides such as SiH(β1) and SiH2(β2), precursors for the formation of the etching product, SiH4(g). Hence, the ability to stabilize the intermediate hydrides helps to promote the etching reaction, which dramatically decreases when the temperature increases.4-6 The decisive importance of atomic hydrogen in diamond synthesis was already emphasized.7 Hence the process involves a strong etching of the graphitic carbon species on behalf of diamond growth and stabilizes surface sp3 hybridization. Thus, atomic hydrogen is absolutely required to grow high-quality diamond. However, the presence of hydrogen radicals has many secondary effects, particularly on nucleation. Kim et al.8 reported that the erosion by atomic hydrogen has an annihilation effect on the nucleation sites of the silicon surface. At a filament temperature of 2273 K, they observed a strong decrease of the diamond nucleation density, which is divided by 30 compared to a noneroded silicon sample. On the other hand, many speculations invoke the role of defects on the surface to generate nucleation sites and to enhance nucleation density. Hence the role of some terraces had been pointed out9 while an STM investigation allowed one to distinguish several kinds of surface defects produced during the silicon etching.10 According to these authors, the deeper defects may act as preferential sites for diamond nucleation. However, generally an accurate structural and chemical description of these sites is still lacking. Furthermore, the real participation of the atomic hydrogen to * Corresponding author. E-mail: [email protected].

the diamond nucleation mechanism is not yet well-understood. It is clear that the nucleation mechanism must involve a set of chemical reactions with removal of molecular hydrogen through the action of gaseous hydrogen. In a recent theorical study, Mahalingam et al.11 proposed a CH3-based nucleation process including radical-radical surface reactions that are both energetically and entropically promoted. Furthermore, they estimated a critical diamond cluster size in the nanometer range. The influence of the nature of the sites was not investigated however. A thorough knowledge of the etching process under CVD conditions is first required in order to better explain the diamond nucleation mechanisms. This is the main topic of the present paper which precisely studies the physical and chemical mechanisms occurring during the silicon etching process. In particular, we emphasize the influence of the methane content in the interaction with atomic hydrogen species. It is found that the presence of hydrocarbons with hydrogen strongly suppresses the silicon etching. This is explained by the steep decrease of the atomic hydrogen in the gas phase. Nevertheless a small amount of carbon speeds out the etching process. This is accounted for by an inhibition of the bulk diffusion of hydrogen owing to preferential carbon adsorption. The size, depth, and angular distributions of the etch pits were studied both by scanning electron microscopy (SEM) and atomic force microscopy (AFM) observations, whereas the nature of deposited carbon was controlled by Auger electron spectroscopy (AES). Experimental Section The samples are B-doped (7-20 Ω cm) Si(111) wafer. After a mechanical polishing, the damaged overlayer and the native oxide were removed by a chemical etching.12

S1089-5647(98)00974-2 CCC: $15.00 © 1998 American Chemical Society Published on Web 06/05/1998

Silicon Etching during the HFCVD Diamond Growth After an ultrasonic cleaning in ethanol, the silicon samples were introduced in a synthesis chamber connected to an analysis chamber where surface analysis by electron spectroscopies could be performed (AES). The experimental apparatus has been elsewhere described.13 Diamond was deposited by the hot filament chemical vapor deposition (HFCVD) method, where a mixture of hydrogen and methane is partially decomposed into radicals by hot filaments. For the set of experiments further described, rhenium was preferred to tungsten as filaments because of their better chemical and mechanical stability. The inertness of rhenium to carbon avoids a precarburization step,14-15 prevents a poisoning of the surface of the filaments, and allows one to reach the steady-state concentration of radicals as soon as the CVD conditions are implemented, whatever the methane concentration. The substrate temperature, measured by a chromel-alumel thermocouple, was 923 ( 5 K. The total pressure was 3000 Pa and the gas flow was 200 L/min. Under these conditions and carried out on a substrate that is not seeded with carbon residues, the diamond nucleation was a very slow process, and we checked that the concentration of diamond nuclei after 60 min of carbon deposition is vanishingly small. To study the etching process, the methane content was allowed to vary within 0-1%. Moreover, to modify the radical hydrogen concentration in the gas phase, we worked with two values of the activation power dissipated in the filaments. These power values, 140 and 170 W, correspond to Re filament temperatures of 2198 and 2273 K, respectively. They were measured by a bichromatic optical pyrometer working at λ1 ) 0.85 µm and λ2 ) 1.27 µm, respectively. The dissociation rate of molecular hydrogen was estimated from the Re filament temperature by taking into account the thermochemical equilibrium between dihydrogen and monohydrogen.16-17 SEM observations were performed on a JEOL JSM 840 microscope, and the images were recorded with an acceleration voltage of 15 or 20 kV. In our experimental conditions, the lateral resolution was estimated to 20 nm. SEM micrographs were digitalized by an optical scanner. Image processing was performed by the use of Khoros freeware. After a binarization step, the etched surface ratio was determined by a threshold of the gray levels. AFM observations were carried out in the tapping mode on a Nanoscope III microscope. In this configuration, where a silicon cantilever vibrated near its resonance frequency (290420 khz), the interaction of the cantilever with the diamond islands was minimized. The effect of the force gradient on the vibration amplitude was measured via an optical detection. By the use of an optimized feedback loop, topographic images of the sample surface were recorded. An exploitation of the AFM line scans led to the measurements of the in-plane and in-depth dimensions of the etched features. The optimum AFM lateral resolution was estimated to 10-15 nm, mainly limited by the tip radius of curvature of 5-10 nm.18 Images with larger errors in the estimation of the lateral dimensions were systematically discarded. We checked the good agreement between the size distributions obtained by AFM and SEM. The convolution effect of the tip was accounted for. In the tapping mode, the silicon tip had a cone half-angle of 25° and the cantilever had a direction that makes an angle, close to 10°, with the horizontal. Results To better understand the mechanisms leading to the etching process of the Si(111) surface during HFCVD synthesis, depositions of 60 min were performed with different methane content in the gas mixture: 0, 0.1, 0.25, 0.5, and 1%. In such

J. Phys. Chem. B, Vol. 102, No. 25, 1998 4857 a time and in the absence of diamond residues left by the substrate pretreatment, the density of diamond islands remains small and never exceeds 106 nuclei/cm2. The etching figures on the substrate are imaged and measured by SEM and AFM observations. Typical AFM images are presented in Figure 1. The smallest etch pits counted have a lateral size of 10-15 nm limited by the best AFM estimated lateral resolution. The densities of the etched pits (number per cm2), determined from both AFM and SEM images, are in excellent agreement although the precision on the determination of the density drops with the methane concentration. They were calculated by taking into account the same surface area (10 µm × 10 µm). The evolution of the density of etched pits with increasing methane content is displayed in Figure 2 for a filament power of 140 W. We note a significant increase of the feature density by ca. 25%, 8.6 × 107 to 1.05 × 108/cm2, when a weak methane content is added to pure hydrogen. Hence, the silicon etching is enhanced by the addition of methane in the range 0.1-0.25%. Further, the etching process is strongly affected by higher methane content, the hole density dropping to 5 × 107/cm2 at 0.5% and beyond 2 × 107/cm2 at 1% of methane content. The image analysis of the AFM pictures allows to calculate the ratio of the etched surface to the total surface. This ratio is plotted versus the methane content on the Figure 3, for the two activation powers of the filaments. At 140 W, the methane content values were 0, 0.1, 0.25, 0.5 and 1% of the gas mixture. At 170 W, the considered methane content were 0, 0.25 and 0.75%. Whatever the activation power, a similar behavior of the etched surface is observed with a maximum for a methane content within 0.1-0.25% and subsequently a strong decrease when the methane content increases. These observations are quite in line with the results concerning the density of etch pits. Size distributions of the erosion features are extracted from the AFM 3D images. The normalized distributions are presented in Figure 4 (for methane content of 0, 0.1, and 0.25% and a filament power of 140 W). More than 100 etch pits spread on several images have been counted for a given methane concentration. The addition of methane leads to narrower distributions, which correspond to a more homogeneous repartition in size of the etch pits. Moreover, we note a significant decrease of the mean size dm while the methane content is increasing. dm is quoted to 436, 329, and 269 nm for 0, 0.1, and 0.25% methane contents, respectively. Furthermore, the mean depth zm is quoted to 17, 11, and 5 nm for 0, 0.1, and 0.25% methane contents, respectively. The statistics becomes so poor above 0.25% that the size histogram cannot confidently be reported. From the surface S and the mean depth zm of the etched pits, an etched volume S × zm can be semiquantitatively estimated, owing to the error in the determination of these parameters. The evolution with the methane content is quite in agreement with the variations of the etched surface and the density previously reported (Figure 5). It again reveals the effect of a weak methane content in the silicon etching. For example, with only 0.1% of hydrocarbon addition to hydrogen, the etched volume is now varied by ca. 60%. The etching process is also promoted by an increase of the filament power (Figure 3). Without methane, both larger and deeper etching holes are obtained when the filament power is increased from 140 to 170 W. As it could be seen on the size histograms of Figure 6, the extracted mean size dm rises from 436 to 521 nm, while the mean depth zm rises from 17 to 30 nm at 140 and 170 W, respectively. These observations are in line with a larger hydrogen dissociation rate with the filament temperature. Thus, at 140 W, the filament temperature is

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Figure 1. AFM images of HFCVD samples as a function of the methane content. a, 0%; b, 0.1%; c, 0.25%; d, 0.5%; e, 1%. The images are recorded in the tapping mode, and the scan size is 10 µm × 10 µm, except image a, which is 30 µm × 30 µm. The white features correspond to surface impurities.

measured to 2198 K and the hydrogen dissociation rate is estimated to 7.6%.16-17 When the activation power is increased to 170 W, the filament temperature reaches 2273 K and the dissociated hydrogen represents now 16.5% of the molecular hydrogen content in the gas mixture. Both the kinetics on the filament and many complex recombination reactions in the gas phase between the filaments and the substrate smear out these values, which are probably overestimated, but we believe the predicted trend is correct. Hence, the enhancement of the concentration of the activated hydrogen just above the surface gives rise to a stronger etching process. The knowledge of both the lateral and depth dimensions, extracted from AFM scan lines, is used to derive R, the direction angle of the etching, and thus can give insight to the mechanism. This angle is estimated to be tgR ) d/2z where d and z are the lateral and the depth dimensions of each individual etch pit,

respectively. We obtain a narrow angle distribution with a mean value centered around 55.4° (Figure 7). It is necessary to account for the convolution due to the geometry of the AFM tip to evaluate the true profile of the etch pit. Assuming that the walls of the pit make a real angle β with respect to the normal of the Si(111) surface, the measured angle R of the AFM scan line will be close to 35°, if β is smaller than 35°. This angle is the addition of the horizontal deviation of the cantilever (10°) and the tip half-cone angle (25°). Now if the β value is higher than 35°, the convolution effect will be weakened. Therefore, we believe that the convolution effect is weak with a measured angle of ca. 55°. This experimental value is in good agreement with a process of silicon etching occurring along (100) directions.19-20 Indeed, the (100) and (111) planes intersects with an angle of 54.6° (Figure 8). It should be noted that this angular distribution is the distribution averaged from

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Figure 2. Evolution of the density of the etch pits versus the methane content. The filament activation power is 140 W. Errors bars are indicated in the plot. Square, SEM micrograph; triangle up, AFM image.

Figure 3. Evolution of the etched surface S versus the methane content for two values of the filament activation power (AFM images). Full square, 140 W; full circle, 170 W.

different methane contents and that perceptible deviations to this mean angle are only observed when Si(111) is etched by pure atomic hydrogen. Discussion The interaction of atomic hydrogen with Si(111) has been manyfold studied, and the results strongly emphasize the occurrence of strongly bound SiH and SiH2 states in equilibrium with weakly bound mobile subsurface hydrogen. The SiH2 surface concentration governs the roughening and the etching of the silicon surface through the formation of a SiH3 complex, which is the limiting rate of the process.1,3,21-22 It is further expected that a rise of temperature will drop the rate of silane evolution, which is explained by the decrease of SiH2 states.1 We found quite effectively a decrease of the silicon etching when increasing the temperature.23 Similar studies are, however, still lacking when a small amount of carbon is added to the gas phase, which is the case of the conditions of the diamond CVD growth. Thereupon, it is well-established that the diamond nucleation is preceded by a rapid formation of a silicon carbide overlayer.24-26 Thus, the silicon surface is rapidly and progressively saturated with carbon. Experimentally, the appearance of the carbide phase is clearly observed on the Si LVV AES spectra (Figure 9), where the kinetic energy of the main minimum shifts from 90.4 to 85.0 eV. In addition, the C KVV line (not shown here) displays the characteristic signature of

Figure 4. Lateral (in-plane) and in-depth histograms deduced from measurements on AFM scan lines for a filament activation power of 140 W. The mean values dm and zm are indicated by arrows and are numerically reported in the text. In-plane distributions: a, 0%; c, 0.1%; e, 0.25%. In-depth distributions: b, 0%; d, 0.1%; f, 0.25%.

Figure 5. Evolution of the estimated etched volume V ) S‚zm versus the methane content. Errors bars are indicated.

silicon carbide. This silicon carbide layer of several nanometers thick is formed during the first minutes of the deposition

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Figure 8. Scheme of the intersect of (111) and (100) planes in the silicon crystal structure. Dark circle, Si(111) plane, bright circles, Si(100) plane.

Figure 6. Effect of the activation power on the size histograms, with pure hydrogen. Mean values, given numerically in the text, are indicated by arrows. Lateral size: a, 140 W; b, 170 W. In-depth: c, 140 W; d, 170 W.

Figure 9. AES Si LVV spectra recorded in the derivative mode. The energy positions indicated by dot lines correspond to the main minima of the silicon carbide and the elemental silicon contributions, at 85 and 90.4 eV, respectively. A, clean Si(111) surface; B, after subsequent 60 min of HFCVD diamond growth at a methane content of 0.5%.

Figure 7. Angular distribution of the etch pits. The mean value is indicated by an arrow. This distribution includes around 300 pits on Si(111) etched with a methane content varying within 0-0.5%.

process.25 Moreover, previous in-situ XPD measurements27 have shown that an epitaxied β-SiC(111) is formed on a Si(111) surface. Therefore, two major cases are examined in our model. The first one corresponds to weak methane contents where the silicon carbide could not fully cover the surface. Here, the hydrogen diffusion into the silicon bulk must be considered.28 In the second case, for higher methane content, the rapid formation of a complete carbide layer prevents the hydrogen

diffusion into the bulk. This affirmation is supported by theoretical calculations that predicted that the hydrogen diffusion into a silicon carbide layer is energetically much more difficult than into silicon.29 Albeit there are presently no experimental evidence of these predictions, this is a crucial point to consider when addressing the interaction of silicon with atomic hydrogen in the presence of carbon. Hence we develop in the Appendix part a model that enables one to calculate the rate of silicon etching through silane removal in the two limiting cases where hydrogen bulk diffusion is accounted for or not. The main interest of this model is that, whatever the assumption on the occurrence of hydrogen bulk diffusion, it predicts a maximum of silicon etching in the presence of carbon in the gas atmosphere. This maximum is shifted toward small carbon concentration when the hydrogen bulk diffusion into silicon is accounted for (Figure 10). This is in close agreement with our results that outline the occurrence of such a maximum at about 0.1-0.25% of methane content in the gas phase. We will discuss the opportunity of the main assumptions of this model and the significance of the results on silicon etching and finally we will point out from these results some consequences on the diamond nucleation. Let us recall the main assumptions concerning our modelization of the interface chemistry between a gas mixture of activated methane and hydrogen and a silicon surface. First it

Silicon Etching during the HFCVD Diamond Growth

Figure 10. Evolution of the silicon removal G as a function of the methane content. Full circle, without accounting for the hydrogen diffusion into the silicon bulk; full square, accounting for hydrogen diffusion into the silicon bulk.

Figure 11. Evolution of the carbon surface coverage with the methane content according to (full circle) kinetic theory of gases (eqs 2-5) and (full square) the data reported by Frenklach et al.34

is of interest to consider the gas phase and the evolution of the various active species while varying the methane concentration. When working under the CVD conditions of diamond growth, this point is now well-documented, both experimentally and theoretically. Hence optical and mass spectrometry measurements have provided reliable data on the concentration of the active species in the gas phase above the silicon substrate.30-32 They also agree to predict that the methyl radical CH3 is the active species involved in the diamond growth. Therefore we use the data by Hsu31 to estimate the concentration of each active species (H‚, H2, CH4, CH3, C2H2) as their deposition conditions are very close to our experimental conditions. Equilibration in the gas phase occurs between the hydrogen and the methyl radicals close to the surface, according to chemical equilibrium3 and to eq 5 reported in the Appendix.31-34 The next step is the progressive and far rapid coverage of the silicon surface by carbon. This is described by a Langmuir isotherm, accounting for the limited amount of adsorption sites. We check by two ways that the calculations of the carbon coverage as a function of the methane content yield a good agreement. The first one uses the data by Frenklach and Wang,34 and the other one uses the kinetic theory of gases (Figure 11). Finally the silicon etching occurs via a mechanism that has been thoroughly described in the literature through the interaction of subsurface weakly bound hydrogen of concentration n0 in equilibrium with SiH and SiH2 adsorbed species, precursor of the silane gas-

J. Phys. Chem. B, Vol. 102, No. 25, 1998 4861 phase evolution, and diffusing hydrogen. The calculation of this steady concentration of subsurface atomic hydrogen n0 as a function of the carbon content is of great importance on the silicon etching, as evidenced by eq 1. The occurrence of a maximum at low carbon content can be explained by the balance between two effects that operate in reverse directions on n0. The first one is a rapid carbon coverage that inhibits the bulk hydrogen diffusion and thus contributes to increase the subsurface hydrogen concentration. The second one is the progressive depletion of the gas-phase atomic hydrogen concentration governed by the chemical equilibrium3 that leads to a decrease of n0. In addition to this semiquantitative result on the silicon etching, the AFM studies provide additional information on the nature of the sites left by the silicon etching. Thus we observe a narrow angular distribution very close to the Si(100) direction. This well agrees with the conclusions of an in-situ STM study of the silicon etching.6 The authors calculated the etching probability p of different silicon surface orientations. They found that p of a silicon atom issued from a Si(100) surface is about three times larger than that of an atom issued from a Si(111) surface. They explained their experimental results by strain considerations. Hence the stability of the SiH2 complex that is the precursor to Si etching is larger on a Si(100) surface where the silicon atoms have two back-bonds with subsurface Si atoms. By contrast in the case of a Si(111) surface, the insertion of an hydrogen atom into one of the three back-bonds leads to a strong strain, which prevents the dihydride formation.5 This difference can explain why the silicon etching mainly and selectively occurs along Si(100) planes. What are the implications to the diamond nucleation of such changes in the surface morphology? The influence of silicon etching by hydrogen on the diamond nucleation has been the subject of few studies.10,35 Chemical treatment by HF:HNO3 or by H2 plasma were carried out on silicon samples that were preliminary mechanically or ultrasonically scratched. The results generally emphasize the strong annihilation of the nucleation sites with long preliminar hydrogen treatment, although short hydrogen treatments may enhance diamond nucleation. The annihilation of the nucleation sites can be attributed to the generation of Si(100) planes onto which the diamond nucleation is inhibited. Hence we observe a very weak and very long nucleation process on a pristine Si(100) surface compared to a pristine Si(111) surface, both surfaces being free from carbon residues.36 In the same trend a correlation has been found between the diamond nucleation on virgin silicon surfaces and the density of dangling bonds of the silicon surface,37 so that the nucleation on Si(111) is much more effective than that on Si(100).38 By contrast, the increase of diamond nucleation with short exposure to hydrogen can be assigned to a temporary roughening of the silicon surface, which points out a dense silicon surface. The effect of the presence of carbon was studied by Kim et al.8 For a methane content of 0.3%, a strong etching leads to an annihilation of the diamond nucleation whereas, with 1% methane the etching process is much weaker. This is consistent with our conclusions if we consider the fast drop of the etching process when increasing the methane content. The conclusion to draw for further studies is that etching by atomic hydrogen is detrimental to diamond nucleation as it generally promotes low-density silicon surfaces. To overcome this effect, it would be preferable to start the nucleation process at high methane content as both the sursaturation of carbon is increased and the etching process is inhibited. It should be noted that these results are at variance with the interpretation of a size

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effect on diamond nucleation by Yugo et al.39 Interpreting STM images, they claimed that the proper sites for diamond nucleation are surface defects of comparable size and shape to critical diamond nuclei. In standard CVD experimental conditions, they estimated this critical size to several nanometers.10,39 Although hard to evidence by AFM, it is however expected that the density of such etch pits will be larger at high atomic hydrogen concentration, whereas a drop of the diamond nucleation is observed. We are presently performing diamond CVD at longer deposition times in order to understand the role of the etching defects in the diamond nucleation mechanism. Conclusion In conclusion, we evidence in this study the strong effect of the methane content on the etching process of a Si(111) surface. The silicon etching is slightly exalted with addition of 0.10.25% of methane in the feed gas and depleted with larger addition of carbon. This etching occurs easily along Si(100) directions. This is explained by the better stability of the precursor SiH2 to silane removal on (100) than on (111) planes. A modelization of the process points out the balance on the surface hydrogen concentration when increasing the methane content between the drop of the atomic hydrogen concentration in the gas phase and the inhibition of hydrogen bulk diffusion into silicon. This explains the occurrence of a maximum of silicon etching. Possible consequences on the diamond nucleation process are then put forward. It is thus expected that the etching of the silicon is generally detrimental as generating lowdensity silicon surface such as Si(100) where diamond nucleation is inhibited. Appendix Etching of Si(111) in the Presence of a HydrogenHydrocarbon Mixture. Assuming that steady-state conditions are quickly fulfilled, the formation rate of silane SiH4 has been expressed as1

d[SiH4]/dt ) G ) 4‚k3‚Γ‚Ns‚θSi‚n0

(1)

where k3 is the kinetic constant of the limiting rate reaction

SiH2(ads) + H(ads) f SiH3(ads)

(1a)

which is a Langmuir-Hinshelwood reaction affecting both (111) and (100) planes.3,5 Hence the SiH2(ads) species are formed by filling dangling surface bonds. Further silane evolution occurs through subsequent hydrogen incorporation

SiH3(ads) + H(ads) f SiH4(g)

(2a)

which is being considered as quite fast. Throughout the text, (ads) and (g) will mean the radical in the adsorbed state and in the gas phase, respectively. A Si(111) surface exposed to hydrogen exhibits both SiH(ads) and SiH2(ads) bound states supplied by weakly bound and mobile subsurface hydrogen states of steady concentration n0.21-22,40 In eq 1, Γ is the fraction of the surface covered by SiH2(ads) relative to SiH(ads), Ns is the total number of silicon surface sites and θSi is the fraction of surface exposing silicon. Γ can be extrapolated from the STM studies by Boland using an enthalpy activation of 14.5 kJ/mol for the transition between the SiH(ads) and SiH2(ads) states.5 Owing to the high hydrogen coverage, it is expected that the surface reconstruction does not occur even at 923 K. Therefore the formation of stable SiH3(ads) states on a reconstructed surface is neglected.41 Furthermore θSi drops with

the increase of the carbon concentration. We neglect the change in the amount of surface sites due to roughening when carbon coverage occurs. Despite the large difference of the lattice constants between silicon carbide and silicon (38%), we preliminary showed that the silicon carbon layer grown in the present conditions is too thin (1-2 nm) to generate by itself a strong roughening of the surface leading to a significative change in the density of surface sites.42 Therefore Ns is assumed to be constant. Finally the thermal accommodation coefficients for the sticking of H‚ as well as CH3‚ radicals are not known and will be neglected. Accounting for this effect would result in an increase of the surface temperature. Within these assumptions the increase of the silane evolution with a small carbon content observed in the experimental study can only be accounted for according to eq 1 by an increase of the steady weakly bound hydrogen concentration n0. Whereas hydrogen diffusion across silicon is well-known, an inhibition of the hydrogen bulk diffusion occurs with carbon adsorption. We assume that the surface coverage by carbon species as a function of the carbon gas-phase concentration can be expressed by a Langmuir isotherm, accounting for the limited amount of surface sites onto which carbon can be adsorbed. For convenience purposes, we assume that methyl radical is the unique carbon species involved in the process. This point has been often checked in the literature.30-32 Thus

θC ) k′‚[CH3]/{1 + k′‚[CH3]}

(2)

Therefore, as θC + θSi ) 1

θSi ) 1/(1 + k′ [CH3])

(3)

where k′ is the Langmuir coefficient rate. We extract it from the kinetic theory of gases,

k′ ) [γc/(1 - (γc/2))]‚[8‚kB‚T/(π‚mc)]1/2‚(σs‚Ns)/4 (4) where kB is the Boltzmann constant, σs is the collisionnal cross section of the surface site, and mc is the molecular mass of the gaseous species attacking the surface. γc is the surface reaction probability which is obviously very large when dealing with radicals impinging the surface. Moreover the reaction is highly exothermic.11,43 Owing to the occurrence of a forced flux of methyl radicals, the factor [γc/ (1 - (γc/2)] instead of γc accounts to a first approximation for the non-Maxwellian distribution of the velocities when γc is large.44 Furthermore, it is known from comprehensive studies of the chemistry of the gas phase in the course of the diamond deposition that a quasi-equilibrium of the concentration of the hydrogen and hydrocarbon radicals is reached in the gas-phase just above the surface according to31-34

K(T) ) [CH3]‚[H2]/[CH4]‚[H‚]

(5)

where T is a gas temperature that ranges within the temperature of the filaments Tf and the temperature of the substrate Ts (Ts e T , Tf) and K(T) is the constant corresponding to the equilibrium

CH3(g) + H2(g) T CH4(g) + H(g)

(3a)

Figure 11 reports the variation of the carbon coverage with methane content using eqs 2-5. They are compared with the data by Frenflach et al. on the surface adsorption of a methyl radical.34 Despite the many assumptions the agreement is satisfactory and underlines a rapid surface coverage by carbon.

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The rate of hydrogen adsorption dn/dt can be expressed as

dn/dt ) [η‚I0] - [4‚k3‚Γ‚θSi‚n] - [2‚kH‚θSi‚(1 - Γ)‚n] [θSi‚D‚(δC/δz)] (6) where the first term of the right side of this equation is the hydrogen adsorption rate, the second one the rate of hydrogen removal through silane formation from SiH2(ads) according to chemical equations 1a and 2a, the third one is the rate of hydrogen removal through the recombination of surface hydrogen (eq 4a) according to a Langmuir-Hinshelwood mechanism

SiH(ads) + H(ads) f Si + H2(g)

(4a)

and the last term accounts for hydrogen diffusion into the silicon bulk. The flux of radical hydrogen can be expressed as η‚I0, where η is the overall sticking coefficient of hydrogen radicals and I0 is the hydrogen radical flux impinging the surface. Assuming as well just above the substrate a non-Maxwellian distribution of the velocity of the hydrogen radicals due to the forced flow, the kinetic theory of gases derives the rate of impingement of hydrogen radicals I0

I0 ) (8‚kB‚T/(π‚mH))1/2‚Ns‚[H(g)]/4

(7)

where mH is the molecular weight of hydrogen radicals. The sticking probability η is strongly dependent on the nature of the sites, so that it can be expressed as

η ) (ηSi‚σSiθSi/(1 - (ηSi/2))) + (ηC‚σC‚θC/(1 - (ηC/2))) (8) where ηSi and ηSi are the sticking probabilities of hydrogen on carbon and silicon sites, respectively, and σSi and σC the collisionnal cross sections of hydrogen on the silicon and carbon surface sites, respectively. The adsorption rate can therefore be expressed using both expressions 7 and 8. In the third term of the right side of the differential equation 5, kH is the kinetic rate of hydrogen recombination on silicon. The hydrogen recombination on carbon is neglected as, instead of silicon, it proceeds by direct reactivity of gaseous hydrogen (Eley-Rideal mechanism).45 Thus the rate is expected to be much lower, both considering the entropic and the energetic terms.11 In the last diffusion term D is the diffusion of hydrogen of bulk concentration C. It is assumed that this diffusion term cancels when carbon fully covers the surface. Calculations had pointed out that carbon diffusion into silicon carbide requires a much higher activation energy.29 Therefore we will assume a carbon diffusion occurring across the surface silicon sites only. We wish to develop the expression 6 into two limiting cases: (1) First, it is assumed that the diffusion is negligible, a case that occurs when the carbon coverage is high (θSi , θC). A stationary hydrogen concentration n0 can be analytically derived, with dn/dt ) 0. Then

η‚I0 ) n0‚θSi‚{[4‚k3‚Γ] + [2‚kH‚(1 - Γ)]}

(9)

Accounting for expressions 2-4 and 6-8, we can derive a steady concentration of subsurface hydrogen n0 in the form

n0 ) (a + b‚[CH3])‚[H]

(10)

where a and b are constants. Putting n0 into equation 1, we obtain the silicon etching rate

G ) 4‚k3‚Γ‚Ns‚θSi‚n0 ) (a + b‚[CH3])‚[H]/(a′ + b′‚[CH3]) (11) where a′ and b′ are other constants. Whatever the carbon coverage, the rate of silane evolution is a constant that both depends on the activated hydrogen and the carbon concentrations in the gas phase. (2) Now the diffusion rate is not negligible, and this is the case where the silicon coverage is important. The hydrogen bulk diffusion can be limited in rate by molecular recombination or trapping on deep or shallow defects of silicon.28 However, owing to the low doping level of the sample and to the occurrence of an hydrogen forced flux, the molecular recombination or trapping on defects can be neglected. Then the resolution of the diffusion equation according to eq 6 yields46

n(t) ) n0‚{1 - exp(J2‚t/D‚θSi‚h1/3)‚erfc {J‚[t/ (D‚θSi‚h1/3)]1/2}} (12) where

J ) θSi‚n0-1/2‚{[4‚k3‚Γ] + [2‚kH‚(1 - Γ)]}

(13)

is the first-order rate constant of the competitive hydrogen surface reaction and h is the solubility coefficient of hydrogen. Finally the amount of silicon etching through silane evolution is

SiH4 )

∫tG‚dt ) 4‚k3‚Γ‚Ns‚θSi‚n0‚{t -

∫texp(J2‚t/D‚θSi‚h1/3)‚erfc [J‚(t/D‚θSi‚h1/3)1/2] dt}

(14)

We report in Figure 10 the variation of the rate of silicon etching G as a function of the methane concentration with values extracted from the literature in the two cases above-described. The following values are used: k3 ) 2.7 × 103‚exp(-7534/ R‚T) (s-1)1; D ) 9.4 × 10-7‚exp (-46 269/R‚T) (m2‚s-1);47 kH ) 6.2 × 106‚exp(-18 835/R‚T) (s-1)1; Γ ) Γ0‚exp (14 500/ R‚T) with Γ0 ) 0.2 as extracted from the (3 × 1) superstructure of Si(100):H5 at 400 K and the enthalpy change reported in ref 1; t ) 3600 s; ηSi ) 0.7 ( 0.1 [1]; ηC ) 1;28 T ) 923 K; R ) 8.31 USI; Ns ) 11.76 × 1018 m-2;37 h ) 2.4 × 1021 exp(181 217/R‚T) m-1;47 σSi ) 3.14 × 10-20 m2/site. Without diffusion, the etching rate increases smoothly up to about 0.75% as mainly due to an increase of the surface concentration of hydrogen. Accounting for the hydrogen diffusion leads to a drop of the silicon etching and to a shift of the maxima toward lower carbon concentrations (0.1-0.2%). Owing to the many assumptions made and the use of data of the literature, we only account for the trend observed by this model, which predicts an optimum etching at low carbon concentration. References and Notes (1) Abrefah, J.; Olander, D. R. Surf. Sci. 1989, 209, 291. (2) Pearton, S. J.; Corbett, J. W.; Shi, T. S. Appl. Phys. 1987, A43, 153. (3) Gates, S. M.; Kunz, R. R.; Greenlief, C. M. Surf. Sci. 1989, 207, 364. (4) Olander, D. R.; Balooch, M.; Abrefah, J.; Siekhaus, W. J. J. Vac. Sci. Technol. 1987, B5, 1404. (5) Boland, J. J. Surf. Sci. 1992, 261, 17. (6) Wei, Y.; Lian, L.; Tsong, I. S. T. Appl. Phys. Lett. 1995, 66, 1818. (7) Spear, K. E. J. Am. Ceram. Soc. 1989, 72, 171. (8) Kim, J. W.; Baik, Y. J.; Eun, K. Y. Diamond Relat. Mater. 1992, 1, 200. (9) Polini R. J. Appl. Phys. 1992, 72, 2517.

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