Comprehensive new insights on the potential use of SiC as plasma-facing materials in future fusion reactors

The performance of silicon carbide as an alternative plasma facing material (PFM) was studied at various irradiation conditions relevant to ion energies and fluxes of a fusion reactor. This analysis involves detailed modeling of subsurface plasma/material interactions, sputtered particle transport above the surface and redeposition, and related changes in material composition and microstructure induced by steady-state and Edge Localized Mode ion fluxes. Transition of a crystalline SiC surface to semi-crystalline and amorphous phases was analyzed based on advanced modeling of DIII-D tokamak experiments where SiC was irradiated in single- and multiple- L-mode and H-mode discharges. This analysis shows that displacement damage, particle deposition/redeposition, and D accumulation on the SiC divertor surface can lead to significant microstructural changes that result in enhanced sputtering erosion in comparison with the original crystalline material. However, the resulting total net erosion rate for a full-coverage, advanced tokamak, SiC coated divertor may well be acceptably low. Moreover, the C sputtering yield from the evolved SiC surface can be seven times lower than from a pure graphite surface; this would imply significantly reduced tritium co-deposition rates in a D-T tokamak reactor, compared with a pure carbon surface. It was also determined that chemical sputtering of both C and Si should not result in any noticeable effect on the net erosion, for attached plasma regimes. Our results thus show encouraging results overall for use of SiC as a PFM in tokamaks.

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Introduction
Acceptable operation of fusion reactors will highly depend on the plasma facing materials (PFMs) response to high heat and particle fluxes during steady-state and transient plasma operation.Degradation of material properties as a consequence of plasma irradiation is one of major obstacles for successful development of a fusion reactor.While analysis of tungsten as a main candidate for PFMs has been a major subject worldwide for several decades, finding an optimum solution for protection of a tokamak divertor as well as first wall surfaces still requires significant investigation and development.Several undesirable and even unacceptable effects can occur in the case of a W coated divertor surface including potential core plasma contamination resulting in plasma disruption [1,2] and damage of nearby components due to high radiation power from the plasma generated during the transient events from the evaporated W divertor surface [3].These potential problems call for further improvements of tungsten material through alloying with various elements, or via changed microstructure, e.g.fiber-reinforced composites [4].Alternatives to tungstenbased PFMs, like liquid metals or non-metallic low-Z refractory materials such as silicon carbide (SiC) based composites [5], are also continuing to attract attention.Advantages of SiC include good thermal properties of bulk crystalline SiC, sublimation (not melting) under high heat fluxes, and significantly lower radiation from Si in comparison with high-Z W. Recent experiments showed better performance of SiC under Edge Localized Mode (ELM)-and disruption-relevant heat loads in comparison with W and other W-alternative PFMs [6].Also, recent modeling predicted reduction of Z eff in the core in the case of SiC walls, compared to graphite, due to enhanced oxygen gettering [7], similar to the siliconization process in previous tokamak experiments.While various studies of SiC irradiated by hydrogen isotopes were conducted in laboratory facilities, there are only a few detailed analyses of SiC in real fusion devices [8,9].Some of the main concerns during reactor operation are material erosion, redeposition, and transport to the plasma core.These processes highly depend on the ability of crystalline material to withstand structural changes due to irradiation.Recent experiments at DIII-D showed more than two times lower gross erosion of crystalline SiC in comparison with amorphous SiC material [8].However, these results were obtained from a single discharge on the original crystalline and amorphous surfaces.Further work showed the results of gross erosion yields of crystalline SiC irradiated at different plasma temperatures during many-discharge operations at DIII-D [9].We used these results as well as results of several laboratory experiments to evaluate physical and chemical sputtering, transport and redeposition, and net erosion of amorphous and crystalline SiC at various irradiation conditions.We compare results for the original amorphous and crystalline surfaces and assess the evolution/transition of crystalline surface to amorphous phases.The effects of plasma conditions on Si and C enrichment on the surface and the resulting changes in erosion dynamics were also analyzed in detail.
Accurate analysis of material erosion, transport, and redeposition in fusion devices requires detailed models of under and above surface processes.We coupled our Monte Carlo ITMC-DYN [10] surface evolution/response and REDEP/WBC [11] erosion/redeposition code package to simulate plasma/particle interaction with materials, particle distribution/accumulation in the subsurface layers and trajectories in the sheath layer above the surface.We analyzed several experiments regarding the physical and chemical erosion of crystalline and amorphous materials and predicted the level of surface amorphization during ion irradiation.Binding energies in crystalline and amorphous SiC were evaluated from various experiments and molecular dynamic simulations.We compare our modeling of Si and C erosion with measurements in DIII-D for a wide range of operating conditions.We also analyzed Si and C near-surface plasma transport and redeposition to identify potential core contamination issues.

Review of coupled models
This work uses models implemented in the REDEP/WBC erosion/redeposition code package coupled with the ITMC-DYN surface response code package.The coupled analysis gives a self-consistent, rigorous result for the evolved steadystate surface composition and microstructure, for the gross and net sputter erosion of a DIII-D Divertor Materials Evaluation System (DiMES) probe SiC surface, and for individual erosion rates of Si and C. DiMES is a removable probe built into the lower divertor.The DiMES probe has undergone many modifications over the years [12,13].
The ITMC-DYN package computes the incident plasma particles interaction with materials and integrates Monte Carlo and deterministic models [10,14].The package includes interatomic potential functions for detailed modeling of ion/atom collisions; models for the energy loss of elastic and inelastic collisions; particles thermal diffusion and segregation processes in near-surface layers, hydrogen isotope trapping, molecular surface recombination and desorption, chemical and physical erosion, and calculation of rate coefficients in multi-component materials based on the dynamic material composition.The collisional processes responsible for target atoms sputtering, mixing, and particles reflection are integrated with detailed models for temperature-dependent processes leading to temporal and spatial material evolution.All models are integrated in a self-consistent manner that allows accurate simulation of real experimental conditions and parameters, such as: multiple and simultaneous ions composition, energies, and fluxes; material characteristics and temperature; and the experimental setup.The integrated models have been extensively benchmarked with both in-house experiments and published data worldwide [10,14,15].The ITMC-DYN package is continuously being upgraded and enhanced for the analysis of PFMs in fusion reactor environments.Recently, treatment of inhomogeneous material structure was added to simulate compounds consisting of strongly and weakly bonded atoms that corresponds to the crystal and amorphous phases.The REDEP/WBC code package (computing impurity transport via 3-D, 3-V, full-kinetic, sub-gyro-orbit, oblique incidence magnetic field sheath, turbulence, and atomic and molecular processes models), e.g.[11], computes the transport of individual sputtered Si and C atoms, and backscattered C atoms, and resulting ion trajectories, particle-by-particle, using Monte Carlo.For this study a Si or C atom is launched from the DiMES surface with velocity vector chosen from the ITMC-DYN distribution corresponding to the incident particle species, i.e.D, C, Si, and particle velocity.The sputtered atoms are ionized per the local temperature-dependent electron impact ionization rate coefficients and per the electron density.Calculations are based on the plasma parameters measured at/near the DiMES surface using Thomson scattering and Langmuir probes, e.g. in [9].The resulting Si and C ion trajectories and charge state evolution are computed in detail, based on Lorentz force motion and full-kinetic collisions with the plasma electrons and ions.A particle history terminates upon redepositing anywhere on the divertor surface or leaving the near-surface region.Sputtering by redeposited particles is computed.Further description of the computational methods and code coupling is given in [16,17] and references within.Figure 1 shows a schematic representation of coupled modeling of material erosion/redeposition, surface modification, and fuel retention/recycling, incorporating fine details of PMIs under the surface and within the sheath region.
The code package's coupling procedure first uses the WBC code to compute incident angles and energies for D and C ions from the plasma impinging on DiMES.Using these WBC outputs, ITMC-DYN computes the respective sputter yields and sputtered atom velocity distributions for Si and C from the SiC surface.ITMC-DYN outputs, in turn, are used in WBC to compute Si and C gross and net sputtered erosion fluxes, based on transport/redeposition to the SiC surface and redeposited ions' incident velocities.Erosion/redeposition metrics are computed in two ways: (1) based on the respective incident species fluxes and ITMC-DYN sputter yields including self-sputtering by redeposited Si and C; and (2) based on tracking of dynamic changes in the material simulated by ITMC-DYN.These two ways assure accurate results by double-checking all calculations.ITMC-DYN computes the time-evolved surface composition and erosion of the SiC surface, considering all species: D and C flux from the plasma, redeposited backscattered C and redeposited sputtered Si and C. Deposited on the surface Si and C and atoms displaced in collisional interactions are treated differently than strongly-bonded crystalline atoms.This allows simulation of mixed (crystalline and amorphous) phases in material formed during irradiation that distinguishes these simulations from other modeling efforts using a binary collision approximation.Also important to the analysis, erosion and redeposition during edge-localized modes (ELMs) due to D + and C 6+ free-streaming ions from the plasma [18] are included; this will be discussed later.In addition to providing insight into past DIII-D SiC experiments, we seek to extrapolate the use of SiC to further experiments in DIII-D, and to future tokamaks.We note numerous variations in DiMES near-surface plasma parameters for the past SiC experiments, and uncertainties in the plasma data.We use a reasonable set of typical parameters for the WBC simulations.We have only found moderate changes in WBC/ITMC-DYN results for a range of studied near-surface plasma temperatures and densities.

Modeling SiC erosion in laboratory ion beam experiments
First, we benchmarked our simulations of SiC sputtering erosion with the results of ion beam experiments on crystalline SiC.While experimental sputtering yields are rather scattered due to different experimental setups and measurement techniques, the average results in the range of D energies of interest (100-1000 eV) show that sputtering yields of SiC are between sputtering yields of crystalline Si and graphite [19,20].Experimental results also show that sputtering yields of SiC by D with energies ⩾100 eV are not affected by material temperature in the range between 300 K and 500 K [20].This indicates that erosion of this material at the considered ion energies and material temperatures are primarily determined by physical sputtering.Indeed, other experiments showed that chemical erosion of C from SiC irradiated by D are 4 times lower on average than from graphite and as low as ∼10 −3 in the case of D with energy 80 eV at 500 K material temperature [21].Silane formation was not detected either in laboratory experiments [20], or in DIII-D experiments [9].This can be explained by preferential chemical reactions in Si-C-H compounds.Statistical thermodynamic calculations showed that silane formation is thermodynamically unfavorable if a significant amount of H is introduced into the Si and C matrix [22].From the above considerations, we calculated SiC erosion based on physical sputtering only and compared our results with the experimental data.A more detailed discussion of the chemical sputtering issue is given in appendix A.
In the simulation of collisional interactions of particles in materials, three important parameters need to be determined: displacement energy E d , surface binding energy E sb , and atom bond energies E b .Many measurements of displacement energies in crystalline SiC showed high variation in the obtained values for the same orientation of a crystal plane [23].Even larger deviations of results were obtained in MD simulations depending on the interatomic potential used and the modeling procedure [24].The closest results to the average experimental values were obtained using MD with Tersoff + ZBL potential and ab initio calculations with the threshold E d of 38 eV for Si sublattice and 15-19 eV for C sublattice [24,25].MD simulations also showed that the energy required to displace C and Si with 50% probability are 26 and 48 eV correspondingly [24].We used these values in our simulation of crystalline material response to irradiation.
Surface binding energies of crystalline SiC were obtained in recent MD calculations for different surface directions and composition [26].Since measurements of surface composition after irradiation of SiC in ion beam experiments showed Si enrichment on the surface, we used surface binding energies for Si-enriched surface of 14 eV for Si and 11 eV for C. Also, we used these energies as displacement energies in the first monolayer.Si and C atoms in crystal form compact structures with a high covalent bond energy of 4.6 eV [27].We used this value as the inelastic energy loss required to break atom bonds in crystalline material.
Various experimental work shows that crystalline material becomes amorphous at a dose of 1-10 displacements per atom, depending on ion-target combination [28,29].Various modeling work shows that the displacement energies decrease by a factor of 2 upon becoming amorphous, dropping as low as 5 eV in a fully-disordered, amorphous-carbon structure [24,30].To take into account changes in material structure during irradiation, we implemented tracking of displaced and deposited/redeposited atoms in our modeling of ion/material interactions.This differentiation of various atoms in the material enables consideration of both crystalline and amorphous phases.We used the following assumptions in these simulations: (a) mixing of two types of bonds (crystalline and amorphous) can exist in the material; (b) an atom has the properties of amorphous material if it is (1) displaced by collisional interactions or (2) deposited from plasma impact (C) or locally redeposited (C and Si).The surface binding energy in amorphous material was taken as 5.5 eV (for both Si and C) based on the latent heat of evaporation for SiC.More details on the simulation of the amorphization process are given in appendix B.
Figure 2 shows the results of calculations of sputtering yields of SiC by D ions at normal incidence.The results were produced with the assumptions of unchanged crystalline properties during irradiation (green line) and simulated surface amorphization due to displacement damage (red line).Surface amorphization results in two times increase in sputtering yields and these values correspond to the average experimental results at low material temperatures [20].More details on sputtering yields, that is calculation results in dependence on angle of incidence and energy can be found in appendix B.
Figure 3 shows our simulations of the evolution of SiC surface amorphization due to 1 keV D irradiation.The dependence of the amorphization process on the ion fluence is found to be similar to the experiments where graphite and crystalline silicon were irradiated by light ions (He) with 30 keV energy [28,29].This dependence on the ion fluence is mainly determined by the energy transferred to recoils in the collisional cascade.The average energy transferred in 1 keV D interactions with SiC is similar to the energy transferred in interactions of 30 keV He ions with crystalline Si within 50 nm target used in the experiment [29].

Impurities transport and local redeposition
Plasma interaction with materials in fusion devices is more complex in comparison with the above laboratory experiments due to magnetic and electric field effects.We added modeling of particle transport in the magnetized plasma above the surface to our simulations of collisional interactions in material and calculated the redeposition fractions.
The simulated test surface is a ∼5 cm diameter DIII-D tokamak DiMES probe with a complete SiC deposited surface.For the WBC modeling we use near-surface magnetic field values from the DiMES-61 experiment [31].Reference near-surface plasma temperatures and density used are T e = T i = 30 eV, n e = 3 × 10 19 m −3 , which are chosen from the range of values cited in [9,31].Particle impingement on the DiMES surface from the plasma is by D + ions and 2% carbon ions.(Sputtering by charge-exchange D 0 atom flux to the surface can be included in the future, if related diagnostics are installed; this is an insignificant erosion source for high-Z materials in the divertor but not necessarily negligible for SiC).The plasma parameters are modeled as constant across the DiMES surface, i.e. in the radial and toroidal directions.Spatial variations have been modeled in the past, when precisely known, and can easily be modeled for analysis of a future experiment, upon diagnostic data and plasma modeling becoming available.However, spatial variation effects are not highly significant for the present purposes of predicting overall SiC erosion performance and surface evolution.
The plasma incident carbon ion charge state distribution used here in WBC is based on edge plasma modeling and data from the DiMES-61 experiment, and from recent SOLPS-ITER plasma code simulations of C ion fluxes to the DIII-D SAS-VW divertor [32].These incident carbon ions are almost entirely in charge states k = 1-4, i.e.C 1+ , C 2+ , C 3+ , and C 4+ (with mode C 2+ ).D + and C k+ ion elevation angles of incidence and incident energies on DiMES are computed using WBC, and are a function of pre-sheath velocities and sheath acceleration.The incident energies are calculated based on pre-sheath D + and C ion flow at the local plasma sound speed-such speed set by the majority D + species-and acceleration of the D + and C k+ ions through a sheath potential of 3kTe.The WBC calculations show incident elevation angles about the same for all plasma D + and C k+ ions, at ∼60 • incidence from the normal, thus fairly oblique.
We analyzed two surface conditions for sputtering erosion/redeposition of Si and C from a DIII-D SiC surface, as discussed in the above ITMC-DYN analysis: crystalline SiC and amorphous SiC.
WBC results are summarized in table 1, in terms of the main erosion/redeposition parameters.The key findings are: (1) The silicon redeposition rate is much higher than the carbon redeposition rate.This is due to lower sputtered Si atom velocities for the higher mass Si, higher electron impact ionization rate coefficients, and resulting shorter Si atom ionization distances from the surface.(2) Similar redeposited energies, angles of incidence, and charge states for both Si and C.
Comparative analysis of results for crystalline and amorphous surfaces shows (1) higher average sputtered energies from crystalline surface than for amorphous SiC, due to more of the sputtering coming from C than D because of high binding energies in material; (2) consequently larger ionization distances and somewhat less redeposition fractions.These results show the importance of detailed, rigorous, coupled computations of the evolving SiC surface.

Modeling of SiC erosion in single-discharge experiments
The results of local impurities transport and redeposition analysis such as characteristics of redeposited particles and redeposition fractions, together with the assumed parameters of the incident D and C ions, were used in the selfconsistent simulations of surface evolution and Si and C gross and net erosion from crystalline and amorphous SiC material.
Initially, we benchmarked our modeling against DIII-D experiments on the effect of a single L-mode discharge on erosion of Si from amorphous and crystalline surfaces [8].The pre-sheath electron temperature and density in these experiments were measured to be 23 eV and 4 × 10 19 m −3 and the deuterium flux was measured as 2.5 × 10 22 m −2 s −1 .The slight difference in plasma parameters in these experiments in comparison with our reference case of 30 eV and 3 × 10 19 m −3 does not lead to any notable difference in the redeposition fraction.In fact, an increase in electron temperature from 23 to 30 eV leads to only ∼10% and ∼20% increase in the impact ionization rates for Si and C atoms, respectively [33,34].On the other hand, the velocities of sputtered Si and C atom increase by ∼10% for such increase in electron temperatures; this tends to compensate for changes in the ionization probability.Also, our transport/redeposition analysis shows that a variation of electron density from 3 × 10 19 m −3 -4 × 10 19 m −3 results only in 5% and 17% increase in the redeposition rates of Si and C, respectively.Integration of these parameter variations only results in 5%-7% differences in the redeposition rates for the two cases considered above.In our simulations of single discharge experiments, we use D ions with a calculated energy of 115 eV and C 1+ , C 2+ , C 3+ , and C 4+ ions with relative concentrations determined in our previous analysis of experiments with DiMES probes [16].We also simulated the redeposited sputtered Si and C ions and the redeposited backscattered C ions.Our analysis shows ∼20% redeposition of backscattered C within the area with 5 cm in diameter (the area of the DiMES sample).We simulated interaction of all deposited and redeposited species described above, with the SiC surface and compared the calculated total gross erosion of Si with measurements.Per appendix A and from the thermodynamic calculations [22], we have assumed that chemical  erosion of Si, for the DIII-D conditions studied, should be negligible.Therefore, the erosion rates are computed due to physical sputtering only.
Figure 4 shows the results of our simulations and experimental results [8] of Si erosion as a function of time from amorphous and crystalline targets.In the simulation of the interactions with the crystalline target, we assumed that the C deposited from plasma and the locally redeposited C and Si have the properties of amorphous material.This assumption gives reasonable agreement with the experimental data.Measurements of Si gross erosion were performed during the first shot on the samples.The first high peaks in the measurements were attributed to loosely-bound Si, possibly in the form of micro-particles [8].The decrease in Si gross erosion at the end of pulse can be explained due to the C enrichment of the surface as a result of the deposition of ∼2% C impurities from plasma.Figure 5 shows the simulation results of C accumulation in a few subsurface monolayers during the discharge at DIII-D.The above results indicate that the effective total sputtering yield of Si (relative to the D flux) from the crystalline sample is ∼4 × 10 −3 .This value, obtained from single L-mode discharge, is 3-4 times smaller than the sputtering yields obtained in another set of experiments at DIII-D where crystalline SiC on DiMES probe was irradiated during multiple L-mode and H-mode discharges [9].Such difference in sputtering yields can be explained by the effect of displacement damage on the surface during long device operation and/or due to high ion energy and fluxes during ELMs that can further lead to surface amorphization and enhances sputtering erosion.We will analyze these effects in the following sections.

Modeling SiC erosion in multi-discharge DIII-D experiments
In the next set of simulations, we analyzed sputtering erosion and redeposition, surface evolution and the effects on net material erosion.A reference case with a near-surface plasma temperature ∼30 eV and density ∼3 × 10 19 m −3 was used; results were compared to a multi-discharge irradiation of a SiC sample at DIII-D [9].[9] indicating that original crystalline SiC surface on DiMES was modified during tokamak operation.We assumed in these simulations that 100% of the chemically sputtered C is promptly ionized and redeposited (see appendix A).Therefore, the chemical erosion yield in the calculation of the total yield of C was not taken into account, which can explain a slightly lower value in comparison with the average of measurements where the chemical gross erosion of C was added to the total yield.The measured chemical erosion yield of C in DIII-D is considered in these simulations conditions to be around 10 −2 [9], which is 10 times higher than that found in ion beam experiments where only D irradiated SiC [21].
We studied the sensitivity of the net sputter erosion rates to the near-surface plasma electron density (n e ). Figure 7 shows the sensitivity study results for both Si and C at steady-state conditions when net erosion of Si and C are the same.For the factor of two density range studied, the net sputter erosion flux increases only slightly (∼15%) with n e .We identify two offsetting effects explaining this: (1) D + and C k+ ion plasma fluxes to the surface scale with n e (linearly in the model used), thereby increasing the gross sputter rates for higher n e ; (2) however, the redeposition rates also increase with n e and this tends to decrease net sputter erosion.We have concluded that the variations of this density parameter, for the range studied, will not make any fundamental difference in our overall conclusions.The WBC/ITMC-DYN sensitivity analysis was also performed for the near-surface plasma temperature in the of range of T e = 20-40 eV.The redeposited ion energies vary linearly with T e , as expected.However, increasing plasma temperatures result in both increasing sputtered particle velocities and increasing the probability of sputtered atoms ionization.When these two effects are taken into account, only little changes in the redeposition rates for Si are expected, due to similar variation of the impact ionization rates [34] and the average sputtered Si velocities, i.e., ∼25% increase of both values with increasing temperature from 20 to 40 eV.The effect of the electron temperature on the C redeposition rate is more pronounced due to a ∼2.5 times larger increase in the impact ionization coefficients compared to an increase of sputtered C velocities within the 20-40 eV range [33].

Surface amorphization during H-mode discharge
Simulations of a DIII-D experiment with a single L-mode discharge on a SiC sample at an electron temperature and density of 23 eV and 4 × 10 19 m −3 respectively (figure 4) indicate that displacement damage during short irradiation times and at moderate electron temperatures will not have a significant effect on Si sputtering.D + ions with 115 eV energy cannot displace Si atoms because the energy transferred in D-Si collisions (maximum transferred energy = 28 eV) is significantly lower than the threshold displacement energy for the Si sublattice (around 38 eV).Only C ions from the plasma, redeposited C and Si, or recoiled C atoms that gain high enough energy in collisional interactions can cause Si displacement.However, the energy required to produce Si vacancy in C-Si collisions is two times higher than in Si-Si or D-Si collisions [23,24].In addition, not all atom displacements lead to defects formation (permanent changing of the original material structure); the number of produced defects is reduced with reducing energy transferred in collisions [24].From this consideration and based on data of single L-mode discharge experiments at DIII-D, we can conclude that higher ion fluence than considered in this 3.6 s discharge is required to produce notable displacement damage leading to significant surface amorphization.More experimental data are required to determine the transition time from crystalline to amorphous surface caused by the low energy D and self-ions irradiation where fast defects recovery needs to be taken into account [24].
If we consider H-mode discharges, displacement damage in crystalline SiC can be caused by both D and C ions during ELMs.For simulating the ELMing plasma, we use the DIII-D data-based ELM free-streaming model of [18], as implemented in WBC [16,32].This model assumes that a flux tube from the plasma pedestal top detaches into the scrape off layer directly to the divertor.The model defines a reference ELM plasma with free-streaming D + and C 6+ flux, both with impinging ion energies of 1000 eV.
The WBC redeposition analysis for ELMing plasma on SiC shows slightly different results in comparison with inter-ELMs plasma operation.These changes are due to the higher sputtered energies of both Si and C atoms from the ELM D + and C 6+ impingement.Also, WBC outputs for backscattered, free-streaming, 1000 eV C 6+ ions needed for the full ITMC-DYN computation in this case show a low redeposition rate (∼0.13) due to high-energy impingement and resulting high average backscatter energy (∼300 eV), and higher ionization mean-free-path (∼14 mm perpendicular to surface).
The results of the redeposition analysis were used in ITMC-DYN simulations of surface evolution during and between ELMs.ELMs with 10 Hz frequency, 1 ms duration, and 10 23 m −2 s −1 D flux [18] were simulated in addition to inter-ELM D and C fluxes, as described above.All details of the plasma species, redeposited Si and C, and backscattered C are taken into account in these simulations.Surface amorphization due to ELMs particles is simulated with the assumption that both deposited and displaced atoms form amorphous material.Figure 8 shows the evolution of net erosion fluxes of Si and C during and between ELMs from the initially crystalline SiC surface.While the net erosion of both Si and C are the same during steady-, inter-ELM operation (due to constant surface composition and plasma parameters), the net erosion of C is about two times higher during ELMs.This can be explained by the changing balance between the amount of deposited, redeposited, and sputtered C. The fraction of deposited C from the plasma remains the same during an ELM.However, the total sputtering yields of both elements are increased during ELMs, leading to increased C erosion and a slight decrease of C concentration on the surface.However, due to the short ELM duration, this does not prevent C accumulation in the first few subsurface layers.The overall result of this analysis is significantly increased peak sputtering fluxes due to ELMs.However, the sputtered fluences depend on ELM parameters and may not be highly increased for moderate values of ELM duty factors.
Our simulations indicate that energetic ions from ELMs can cause significant amorphization of the SiC surface.Due to the much larger range of 1 keV D ions in SiC (around 11 nm) and much more energy transferred in collisions to both Si and C sublattices (in comparison with inter-ELM ions), damage at the 1 dpa level can be produced within a ∼10 nm SiC surface after 10 ELM events.About 3-4 monolayers on the surface can be amorphized even after a few ELM events.In addition to the effect of displacement damage, the high D accumulation on the surface leads to preferential C-D bond formation [22] reducing the probability of original crystalline structure reconstruction.However, preferential C-D bond formation does not lead to a significant C chemical erosion due to the multiple complex bonding in amorphous Si-C-D compound [22].The 10 ELM events with the considered parameters correspond to a total fluence of 10 21 m −2 .A similar dependence of surface amorphization on ion fluence was found in the experiments  where graphite and crystalline silicon were irradiated by light ions (He) with 30 keV energy [28,29].

Effect of C impurities on SiC performance as a PFM
Around 2% of C in the plasma sourced from graphite walls leads to a C accumulation on the SiC surface that results in significant sputtering erosion of C which is comparable with sputtering erosion from the graphite surface, 0.035 against 0.048 for the reference plasma parameters (30 eV and 3 × 10 19 m −3 ).On the other hand, laboratory experiments show that irradiation of SiC by only D ions (at certain ion energies) leads to Si enrichment on the surface [21] that might lead to significant reduction of C erosion.We simulated irradiation of SiC surface with different C impurities content in the plasma to predict the surface evolution and resulting C erosion dynamics.(Simulation of ELM fluxes was not included in this particular analysis.)figures 9(a) and (b) show that eliminating global C impurities leads to five times decrease in C physical sputtering from SiC in comparison with conditions at DIII-D and to seven times decrease compared to C physical sputtering from graphite.Our analysis showed that there is not any notable effect of C chemical sputtering on net erosion, at the attached plasma conditions (see appendix A).All chemically eroded C from the surface is locally redeposited on this surface.Similarly, chemical erosion will not affect the surface composition-all chemically sputtered C from the first monolayer is immediately redeposited on the first monolayer.Simulation results also show that in the graphite-free chamber with negligibly small concentration of C in the plasma, the gross erosion of Si is about two times higher than the gross erosion of C.However, the high gross erosion of Si is compensated by a higher redeposition fraction.
A relatively small effect of the C impurities in the plasma on Si gross erosion was predicted in these simulations.Figure 10 show relative Si concentrations on the surface when steadystate conditions are achieved in the case of 2% of C in plasma (a) and without C in the plasma (b).In a graphite-free chamber, significant surface enrichment in Si occurs due to both preferential C sputtering and ∼2 times higher redeposition of Si than C. Comparison of results on figures 9 and 10 shows a non-linear dependence of Si gross erosion on Si concentration.In the presence of C ions in plasma, the Si concentration is decreased on the surface, which results in reduced Si sputtering by D ions.However, the relatively high energy C ions (200-400 eV) in this case also sputter Si, which leads to increasing the total Si gross erosion and slightly compensates for the concentration effect.
Enrichment of the surface in Si in the absence of C impurities in plasma might lead to silane formation and chemical erosion of Si, which could be high at expected DIII-D material temperatures of ∼500 K [20].However, based on the thermodynamic calculations, the Si-H bonding rate is very low in comparison with Si-Si and C-H bonding in Sirich compounds of Si-C-H [22].The presence of C on the surface prevents chemical erosion of silicon, which explains the absence of silane measurements in various laboratory and DIII-D experiments.

Discussion
The key issues for the use of SiC as a PFM in advanced tokamaks include: (1) sputter and plasma transient erosion; (2) plasma contamination, (3) maintenance of stochiometry and thermo-mechanical properties of SiC; (4) tritium codeposition in sputtered and redeposited carbon from SiC.
Our preliminary WBC/ITMC-DYN analysis of sputtered particles transport/redeposition in the entire divertor area shows that the redeposition fractions of both Si and C on the DIII-D lower divertor could be much higher (approaching unity) than for redeposition only on the very small DiMES probe.This would significantly reduce the net sputter erosion rates on a full SiC coated divertor, i.e. having toroidally symmetric SiC coverage, e.g. in future DEMO, compared to DIII-D.Therefore, a high SiC erosion rate for DIII-D/DiMES does not necessarily translate to a problem for advanced tokamak of full SiC divertor/wall.However, these preliminary simulations are based on the assumption of uniform plasma profiles across the entire divertor.More simulations and experimental data are needed for accurate analysis of sputtered particles transport and redeposition in a larger area than DiMES probe.Preliminary modeling of Si and C transport and impact on plasma contamination in DIII-D experiments were recently performed [7,35].We plan to study this issue via further modeling of future DIII-D experiments capturing all important physics in our integrated WBC/ITMC-DYN package.
Briefly extrapolating our present results to a future D-T, carbon-free tokamak, with toroidally symmetric divertor SiC coverage, we predict a factor of 7 less C erosion from SiC, compared to a pure C surface in such device.Coreplasma contamination should be minimal, based on the predicted very high redeposition rates for full-divertor coverage, (and with higher near-surface electron densities than DIII-D).However, this assessment will require detailed, integrated studies, involving radial Te, Ti, Ne, etc profile effects on redeposition, sputtering by tritium, divertor geometry, ELM parameters for future tokamaks, and related factors.
In general, our integrated simulations with detailed models of sheath and subsurface processes predict good performance of SiC as a PFM at the attached plasma conditions.This includes several factors.High local redeposition of Si results in low impurities transfer to the core.In addition, significantly lower radiation from Si in comparison with high-Z W reduces both the plasma disruption frequency and the damage to the large divertor area and surrounding walls by radiation from secondary, divertor plasma during transient events [36].This needs to be studied in more detail.Surface enrichment in Si in a graphite-free chamber leads to low C erosion in comparison with pure C walls (∼7 times lower) that significantly decreases T co-deposition.Increased Si content on the surface leads to the significant improvement of SiC properties as a PFM.The energetically favorable reactions in Si-C-D compound result in negligible Si chemical erosion and reducing C gross chemical erosion.Net chemical sputtering of C is essentially zero, due to intense redeposition; thus, chemical C erosion should not result in any noticeable effect on the total erosion, for the attached plasma regimes studied.

Conclusion
Our new ITMC-DYN/WBC integrated and detailed analysis of SiC as a potential plasma-facing material indicated attractive properties for future fusion reactors.By considering amorphized surface effects in the modeling of crystalline SiC sputtering in laboratory ion beam and DIII-D experiments we find good agreement with the experimental sputtering yields.Our integrated multi-physics modeling showed that a surface of SiC can be amorphized during a fraction of second at fluxes and ion energies relevant to ELM parameters in DIII-D.Therefore, in the simulation of ion/material interactions during H-mode discharges, details of the original surface structure (crystalline or amorphous) are not critical for the prediction of physical and chemical sputtering erosion.However, it can be important for the analysis of D diffusion and trapping in the bulk of crystalline material.Furthermore, for an attached plasma, it was shown that ∼100% of chemically sputtered particles will be locally redeposited, therefore, net erosion fluxes of Si and C are determined by the physical sputtering, redeposition fractions, and material mixing/composition.Modeling showed that plasma induced effects on the SiC surface in a graphite-free tokamak can lead to significantly better performance of this material in comparison with graphite.Major effects include high sputtered Si redeposition, depletion of C on the surface, and preferential chemical reactions in amorphous Si-C-D layers.Modeling was benchmarked against several experimental results for amorphous and crystalline samples that allowed to determine the effective binding energies in the simulation of these materials.
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Appendix A. Chemical sputtering
There are potential concerns about chemical sputtering of SiC, e.g. as discussed in [8,9].The key issue for SiC net erosion is redeposition of chemically sputtered material.We therefore evaluated erosion/redeposition of both chemically sputtered Si and C from SiC, for DIII-D DiMES conditions.We conclude, for the following reasons, that net chemical erosion of SiC in DIII-D is insignificant compared to physical sputtering, for DIII-D attached-plasma boundary conditions.
For a pure-carbon plasma facing surface chemical sputtering is a very complex process, experimentally shown to involve emission of at least five hydrocarbon molecule species; CH 4 , C 2 H 2 , C 2 H 4 , C 2 H 6 , C 3 H 6 ; and trace C 3 H 8 [37,38].In terms of resulting carbon/hydrocarbon transport in the nearsurface plasma, there are 16 molecules involved, with about 130 relevant atomic and molecular processes, involving both electron and proton impact ionization, dissociation, charge exchange, and recombination [39][40][41].These processes have been included and rigorously studied in previous WBC code package simulations, for carbon surfaces and Si-doped carbon subject to chemical sputtering by hydrogen isotopes [39][40][41].
A critical difference between the transport of chemically and physically sputtered carbon is emission at thermal energies for chemical sputtering, as opposed to ∼100 × higher energies for physical sputtering.Due to the low sputtered velocities for chemical sputtering, plasma transport distances are very close to the surface.This results in high redeposition rates, with precise values depending on the local plasma electron temperature and density, and resulting ionization and other rate coefficients.We can roughly distinguish between a detached plasma regime, Te < 5 eV, with lower electron impact ionization but significant proton impact and electron recombination, and an attached plasma with Te ⩾ 5 eV, with higher electron impact ionization.
For chemical sputtering of carbon from SiC there are molecular dynamics calculations showing similar hydrocarbon species emission as from pure carbon [42].Assuming this to be the case, we can use past WBC analysis for pure C to assess chemical erosion of carbon from SiC.In particular, WBC results for an ITER scenario with carbon surface show nearly 100% redeposition of chemically sputtered carbon on the attached plasma portion of the ITER outer divertor, with 80%-90% redeposition on the detached portion [39].Likewise, WBC simulations of carbon chemical sputtering of the JET Mark II inner divertor show a 95% overall redeposition fraction for a 10 eV near-surface plasma electron temperature [41].Even for a very low 1-2 eV plasma, WBC analysis of the DIII-D D-73 detached plasma DiMES experiment shows 88% chemically sputtered carbon redeposition [39].
Extrapolating these findings to the present study we infer a redeposition rate approaching unity for chemically sputtered carbon from a SiC DiMES probe surface, for the modeled plasma conditions.Therefore, predicted net chemical erosion of carbon from SiC for DIII-D/DiMES is negligible.
Per the above cited WBC analysis, however, there can be non-trivial net erosion of carbon due to chemical sputtering for detached plasma regimes, with still high (∼80%-90%) but less than unity redeposition.In fact, chemical sputtering is likely to be the dominant, normal-plasma (non-transient) erosion process for detached regimes, where physical sputtering tends towards zero.Also, per [39], there are concerns about fusion tokamak tritium retention in chemically eroded and non-locally redeposited carbon, for a detached plasma.It therefore appears less desirable to use SiC for a plasma facing surface exposed to a detached plasma, but SiC might still be acceptable subject to detailed/rigorous analysis.
Turning to the erosion of silicon, the chemical sputtering yield of silicon from SiC is likely much less than for carbon, if not actually zero, e.g. as discussed in [8,9].However, a nonzero rate cannot be ruled out.We proceed under the assumption of some chemical sputtering of Si from SiC, in the form of silane, as variously mentioned in the literature.Regardless of the gross chemical erosion the key issue, as for carbon, is the silicon redeposition fraction.
Data for the molecular species of Si chemical erosion from SiC, at the level of detail for carbon, is not available nor is it critically needed for the present purposes.We therefore performed a WBC simulation looking at the basic features of chemically sputtered Si transport for the reference DiMES conditions.This analysis, while simplified, should be sufficient to define the general effect of chemical sputtering of SiC on net erosion of Si.
Using the assumption of silane-only (SiD 4 ) chemical sputtering of Si, we note similar cross sections for electron impact ionization of silane as for the silicon atom itself [43].Regarding the product ions from silane ionization, we note data for SiH 4 ionization arising from various Si-containing surface materials used in the plasma deposition industry [44].This data shows product ions SiH 3 + , SiH 2 + , SiH + , Si + , and H + .The percentages of these ions depend on electron energy, with the dominant species at the electron temperatures of interest here for DIII-D being SiH 2 + and SiH 3 + .Since for this analysis we are mainly interested in the Si redeposition fraction, we do not need to compute detailed post-ionization reactions of Si-containing molecular ions; instead, we compute the transport of Si in the form of one molecule only, viz.silane ions.Such silane ions should have essentially the same redeposition properties as other Si-containing molecular ions (for the attached plasma conditions).Therefore, using a characteristic thermal emission energy, WBC is used to compute the transport of chemically sputtered SiD 4 from DiMES.The WBC simulation launches SiD 4 molecules at 0.1 eV average energy (using 10 6 histories); with chi square velocity components probability distribution corresponding to thermal emission; determines ionization via Monte Carlo; computes resulting ionized molecule transport due to Lorentz forces and collisions with the plasma; and finally computes Si redeposition to the DiMES probe.(We also performed a sensitivity study finding an insignificant effect of different molecular masses, or moderate changes in rate coefficients, on the conclusions).
The WBC results show 99%, essentially complete, redeposition of chemically sputtered Si to DiMES.As for carbon the reason is clear; the silane molecule mean-free-path for ionization (perpendicular to the surface) is very small, 0.24 mm.This is an order of magnitude less than for physically sputtered Si, see table 1. Silane is therefore ionized well within the magnetic sheath, of width ∼1 mm.Once ionized, the strong sheath electric field causes fast acceleration back to the surface.The simulation computes an average SiD 4 redeposition energy of 44 eV, with average angle of incidence 6 • from the normal, due almost entirely to sheath acceleration.This would not result in high self-sputtering.(Another finding is that the average ionization-to-redeposition time is a very short 3.8 × 10 −8 s, which would limit post-ionization reactions; this supports the computational approach of following silane transport only.) We therefore predict, as with carbon, that net erosion of silicon from SiC by chemical sputtering is negligible for DiMES attached plasma regimes.The net Si erosion rate for chemical sputtering would be some two orders of magnitude less than the gross erosion rate (if any).This issue would need further analysis for low temperature detached plasmas, and/or for future purposes such as detailed analysis of possible spectroscopy diagnostics involving Si-containing molecules.

Appendix B. Surface amorphization and sputtering yields
Several models were developed to simulate the transition from crystalline to amorphous phase in the bulk of materials, starting from a homogeneous model where material becomes amorphous only after reaching certain threshold of defects to a heterogeneous model where amorphization process is initiated at the time of defects formation [45].In general, comparison of various experimental results showed that the dependence of amorphization process on fluence can be represented using logarithmic function.Therefore, we developed the following approach to simulate surface amorphization under ions irradiation in fusion devices.Since the displacement energy is the main characteristic of the material phase evolution in our Monte Carlo simulations, we modified this parameter for displaced atoms during the irradiation using the equation: Sputtering erosion of SiC in our simulations of fusion reactor conditions is mostly determined by D ions at 60 • that is different from the laboratory experiment used for benchmarking (figure 2).Figures B1 (a

Figure 1 .
Figure 1.Coupled modeling of PMI in laboratory and DIII-D experiments with fine details of interactions under the surface and within the sheath region.

Figure 2 .
Figure 2. Comparison of calculated sputtering yields of SiC irradiated by D at normal incidence with the experimental data [20].Calculations were performed assuming unchanged crystalline material properties (green) and amorphized surface due to displacement damage (red).

Figure 3 .
Figure 3. Calculated amorphization of SiC surface due to 1 keV D irradiation.

Figure 4 .
Figure 4. Experimental [8] and calculated Si gross erosion from amorphous and crystalline SiC samples on DiMES probe.The origin of the time scale corresponds to the start of Si 1+ signal measured in the experiments (shots #176 508 and #176 488) and to the beginning of ion irradiation of samples in modeling.

Figure 5 .
Figure 5. Evolution of surface composition during single L-mode discharge on amorphous SiC.

Figure 6 .
Figure 6.Calculated time-dependent erosion of amorphous SiC during H-mode discharge at DIII-D: (a) net erosion of Si and C; (b) total erosion yields (related to gross erosion).Shadowed areas on the right figure show the range of experimental values from multiple discharges with all error bars taken into account [9]: dashed blue and red lines correspond to the minimum and maximum experimental values for C and Si, respectively.

Figure 6
Figure 6 shows (a) net erosion fluxes (gross erosion flux minus redeposition flux) and (b) the total gross erosion yields of Si and C. The calculated total yield for C includes sputtered and backscattered atoms, to be consistent with the spectroscopic measurements of all C 1+ ions.The results indicate the initial stage of accumulation of the deposited C from plasma and the following steady-state conditions achieved at ∼1.5 s (figure 6(a)).The net erosion of Si is slightly decreasing with time due to the increased C concentration in a few subsurface layers.The total predicted yields for the amorphous material (figure 6(b)) are in very good agreement with measurements on the DiMES probe at DIII-D[9] indicating that original crystalline SiC surface on DiMES was modified during tokamak operation.We assumed in these simulations that 100% of the chemically sputtered C is promptly ionized and redeposited (see appendix A).Therefore, the chemical erosion yield in the calculation of the total yield of C was not taken into account, which can explain a slightly lower value in comparison with the average of measurements where the chemical gross erosion of C was added to the total yield.The measured chemical erosion yield of C in DIII-D is considered in these simulations conditions to be around 10 −2[9], which is 10 times higher than that found in ion beam experiments where only D irradiated SiC[21].We studied the sensitivity of the net sputter erosion rates to the near-surface plasma electron density (n e ).Figure7shows the sensitivity study results for both Si and C at steady-state conditions when net erosion of Si and C are the same.For the factor of two density range studied, the net sputter erosion flux increases only slightly (∼15%) with n e .We identify two offsetting effects explaining this: (1) D + and C k+ ion plasma fluxes to the surface scale with n e (linearly in the model used), thereby increasing the gross sputter rates for higher n e ; (2) however, the redeposition rates also increase with n e and this tends to decrease net sputter erosion.We have concluded that the variations of this density parameter, for the range studied, will not make any fundamental difference in our overall conclusions.

Figure 7 .
Figure 7. Effect of variation in near-surface plasma electron density on the net sputtered erosion flux and redeposition fraction of Si and C from SiC DiMES probe in simulations for near-surface electron temperatures of 30 eV.

Figure 8 .
Figure 8. Net erosion of Si and C during (peaks) and between (bottom lines) ELMs.Simulations include evolution of initially crystalline material to surface amorphization.

Figure 9 .
Figure 9. Calculated total yields of Si and C from evolving SiC surface irradiated by D ions with different C impurities content in plasma: (a) 2% and (b) 0%.

Figure 10 .
Figure 10.Steady-state concentration of Si on the surface at various C fractions in plasma: (a) f C = 0.02 and (b) f C = 0.
log 10 (P cr Si + P cr C ) where E d is the displacement energy of atom removed from its original position; E cr d and E am d are the displacement energies of atom in fully crystalline and in fully amorphous material respectively; P cr Si and P cr C are the relative concentrations of Si and C in the crystalline phase.Initial displacement energies for crystalline SiC were taken as E d-Si = 48 eV, E d-C = 26 eV, while for the fully amorphized material, E d-Si = 27 eV, E d-C = 13 eV were used.
) and (b) in appendix B show an additional data for amorphous SiC with the dependence of sputtering yields on D energy and angle of incidence: (a) the results for two D energies used in the simulations of inter and intra ELM regimes; (b) the dependence of sputtering yield on energy and angle showing more pronounced dependence on angle at higher D ions energy.Equal concentrations of Si and C were assumed in these calculations which will change during irradiation due to Si or C enrichment on the surface.

Figure B1 .
Figure B1.Sputtering yield of SiC by D: (a) dependence on angle of incidence for two relevant ion energies; (b) SiC sputtering yields as a function of energy and angle.

Table 1 .
Erosion/redeposition modeling summary for silicon carbide test surface on a DiMES divertor probe in DIII-D.Crystalline SiC and amorphous SiC.For sputtering by plasma D + ions and 2% carbon ions.Values shown for Si and C sputtered from the ∼5 cm diameter probe surface.For Te = 30 eV, ne = 3 × 10 19 m −3 near-surface plasma conditions.WBC/ITMC-DYN coupled code package analysis.
a Average, and where shown (standard deviation), for redeposited ions on DiMES.