APPLIED PHYSICS LETTERS 98, 191905 共2011兲

Thermal conductivity degradation induced by point defects in irradiated silicon carbide Jean-Paul Crocombettea兲 and Laurent Proville DEN/DMN/SRMP, CEA-Saclay, 91191 Gif/Yvette, France

共Received 26 January 2011; accepted 8 April 2011; published online 12 May 2011兲 Irradiations are known to decrease the thermal conductivity of ceramics. This phenomenon is tackled by molecular dynamics simulation of the thermal resistance of point defects in cubic silicon carbide. The additional thermal resistivity due to point defects proves to vary linearly with their concentration. Large variations in the proportionality coefficient with the nature of the defects are observed. From these calculations, an approximate scale for the concentration of vacancies in irradiated SiC is built. © 2011 American Institute of Physics. 关doi:10.1063/1.3589358兴 Energy reactor designers contemplate ceramics and their high melting temperatures for applications affording the up rate of their reactors output. For instance, silicon carbide composites are considered for future fusion or fission nuclear reactors. However, in the nuclear context, a sharp drop of the ceramic thermal conductivity is observed under operation due to neutron irradiation.1 Such a change in material ability to conduct heat is a major concern as it may lead to overheating and thereby to the degradation of the system design. Nuclear energy applications of SiC aim at a temperature range between 200 and 1000 ° C, i.e., above the critical temperature of amorphisation,2 in the saturable regime where the properties under irradiation converge to a well-defined value at high doses. In this regime the microstructure under irradiation consists mainly of a distribution of atomic scale point defects. In particular irradiated SiC does not contain the voids that form at higher temperature in the non saturable regime.3 In such situations, the decrease in thermal conductivity is solely due to the population of defects. The measured thermal resistivity is commonly rationalized as a contribution from defects 共Rdef兲 that adds up to the resistivity of the defect free material. The main contribution to the conductivity decrease, comes from the scattering of phonons by point defects, on which we shall focus in the following. Based on the phonon scattering theory developed by Klemens,4 different expressions may be found in literature for the additional resistivity induced by irradiation defects. In particular one can find various laws for Rdef variation with the defect concentration. For instance, Snead et al.1 distinguish two regimes, with Rdef being proportional to the defect concentration or its square root while Senor et al.5 suggests a variation in Rdef as the cubic root of concentrations. Molecular dynamics 共MD兲 is employed on cubic silicon carbide to calculate at the atomic scale the thermal resistivity induced by different types of point defects, exploring two orders of concentrations. The bonding in SiC is described with an empirical potential of the Tersoff type with an optimal parameterization for the thermal transport.6,7 The simulation box is made of a 50⫻ 10⫻ 10 supercell of the conventional eight atom unit cell of ␤ SiC. The temperature in the box is kept constant at either room temperature or 630 ° C with a Berendsen et al.8 thermostat and the time step of the a兲

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calculation is set to 0.25 fs. The thermal conductivity is calculated with the homogenous nonequilibrium MD 共Ref. 9兲 formalism in which a fictitious external force added to the atomic interactions induces a heat current in the box without a temperature gradient. This heat flux is, to first order, proportional to the thermal conductivity. The additional force parameter is set to 10−6 m−1 共see Crocombette et al.6 for details兲. In order to calculate the thermal resistance, distributions of 30, 100, 300, 1000, or 3000 defects of a given type are introduced at random positions in the simulation cell. We consider vacancies or interstitials of either carbon 共VC兲 or silicon 共VSi兲 type as well as antisites 共AS兲 of both types 共CSi and SiC兲. Concerning interstitials, among the many possible configurations10 we restrict our study to only two, i.e., the silicon TSi and the carbon tetrahedral TC sites, in which the interstitial is surrounded either by four silicon or four carbon atoms, respectively. Four interstitial types are thus considered ITC , ITC , ITSi , and ITSi . A peculiarity arises for silicon C Si C Si vacancies, which are metastable compared to VCCSi close 11 pairs. Hence, the latter are also considered in the present letter. The simulation cell is thermalized for 10 ps at either room temperature or 630 ° C. The fictitious force is then introduced for 225 ps during which the heat flux builds up and stabilizes. From the average of the flux during the last 75 ps of the simulation, one gets the thermal conductivity, which decreases with the concentration of defects as exemplified in Fig. 1 in the cases of ITC , CSi, and VSi. After subtraction of C the resistivity of the perfect crystal 共calculated by the same procedure in the absence of defects兲, the additional thermal resistivity due to the defects unambiguously proves to be proportional to the point defect concentrations so that; Rdef = ␳def关def兴,

共1兲

where 关def兴 stands for the concentrations of defects, expressed as absolute fractions. Equation 共1兲 holds for all types of defects and for different temperatures even for huge amounts of defects 共i.e., up to concentrations of 0.15 at. %兲. The proportionality coefficients ␳def associated with different defect types are named resistivity factors in the following. The fluctuations around such a linear law always remain below 30%. This noticeable uncertainty proves, however, much smaller than the difference between the resistivity factors of different defect types. The values of the resistivity factors are

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© 2011 American Institute of Physics

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Thermal conductivity (W/m/K)

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Appl. Phys. Lett. 98, 191905 共2011兲

J.-P. Crocombette and L. Proville

100 C

10

IC CSi VSi

0.0001

0.001

0.01

0.1

defect concentration FIG. 1. 共Color online兲 Thermal conductivity of cubic silicon carbide as a function of concentrations for various types of defects. Symbols indicate MD values; the lines are obtained from averaging the results according to Eq. 共1兲.

shown in Fig. 2, evidencing the large variations with the type of defect. Going into details of comparison between different defects, it appears, first, that vacancies and interstitials provide one order of magnitude more thermal resistance than ASs. Second, silicon defects are consistently more resistant than carbon defects. Third, vacancies and interstitials of each atomic type have very comparable resistivity factors even if interstitials are, on the whole, slightly less resistant than vacancies. Finally, the resistivity factor of the VCCSi complex is comparable but slightly larger than the one for isolated silicon vacancy. Whereas the variation with temperature apSi , a huge rise of pears limited for most defects, for SiC and ISi resistance with temperature is observed. This uncommon behavior is in fact the consequence of a change in the atomic structure of the defects with temperature, a feature evidenced in our simulations. Few other atomic scale calculations exist on point defects dependant thermal conductivity in ceramics 共see, how-

FIG. 2. 共Color online兲 Thermal resistivity factors calculated for various defects and temperatures. Defect types are vacancies 共v兲, carbon surrounded tetrahedral interstitial ITC, silicon surrounded tetrahedral interstitial ITSi, and ASs. The first two columns are for silicon defects, the last two ones are for carbon defects. Columns 1 and 3 are for room temperature while columns 2 and 4 are for 630 ° C. The last two columns are for the VCCSi cluster at room temperature and 630 ° C.

ever, Refs. 7, 12, and 13兲. We intend a comparison of our results with two earlier works. 共i兲 Kawamura et al.13 calculated the thermal conductivity of 4H SiC at room temperature containing from 0.1% to 1% of vacancies. From their graphical results, we deduce that the additional thermal resistivity is indeed proportional to the concentration of vacancies, in agreement with our findings. Resistivity factors in their calculations prove comparable to ours. 共ii兲 Li et al.7 dealt with the effect of some point defects on the thermal conductivity in ␤-SiC. They did not vary the defect concentrations but considered five different temperatures. Their results show a limited variation with temperature, in agreement with the present letter. Quantitatively the values for resistivity factors deduced from their calculations are larger than those reported here, by 20% to 80%. Moreover they found AS resistivities larger than for other defects, at variance with our results. It is worth noticing that the simulation methods differ as Li et al. used a simulation cell much smaller 共216 atoms兲 and that they introduce only one defect in the cell, which might explain the deviations between the two studies From the theoretical point of view, one can compare our results with the expression given by Snead et al.1 for vacancies

␳vac =

3␲⍀␻D 2kBv2

共2兲

based on the Debye frequency 共␻D兲, average phonon velocity 共v兲, the Boltzmann constant 共kB兲, and the atomic volume 共⍀兲. With the values calculated for the present empirical potential, Eq. 共2兲 leads to a value of 5.5 mK W−1 for the resistivity factor, independently of the nature of the vacancy. Compared to our MD results, this value is of the correct order of magnitude but is about 2 and 3 times larger than our calculated value for the silicon and carbon vacancy, respectively. Resistivity factors enable one to relate quantitatively point defect concentrations and thermal conductivity. Indeed, from the knowledge of concentrations of defects in a material, resistivity factors would allow one to estimate the thermal conductivity. Conversely, it is possible to use the resistivity factors to build an approximate scale for the concentrations of defects in real irradiated silicon carbide samples. A summary of many experimental results on room temperature thermal conductivity of irradiated silicon carbides has been gathered by Snead et al.14 and is reproduced on the left of Fig. 3共a兲, evidencing the large variation in the thermal conductivity with irradiation temperature and dose. The nature and concentrations of defects in these irradiated samples is of course unreachable from these experiments alone. However, one can make the following simplifications to assume that only vacancies defects are contributing to thermal resistance in the samples. We follow here the arguments of Snead et al.1 First, with the experiments being performed at temperatures in the saturable regime, where vacancies are immobile, no void should be created by the irradiations. The irradiation may also create dislocations but their resistance could be safely neglected.1 Moreover, for irradiations being performed at temperatures above the critical temperature for amorphization, interstitials are highly mobile so that their concentration is negligible. If one further neglects the resistivity contribution of clusters of interstitials that also form under irradiation3 as well as the one of ASs

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Appl. Phys. Lett. 98, 191905 共2011兲

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FIG. 3. Left panel 共a兲 summary of experimental measurements of room temperature thermal conductivity after irradiation of SiC at various doses and temperatures; data gathered by Snead et al. 共Ref. 14兲. Reproduced with permission from L. L. Snead, T. Nozawa, Y. Katoh, T.-S. Byun, S. Kondo, and D. A. Petti, J. Nucl. Mater. 371, 329 共2007兲. Copyright © 2007 Elsevier. Right panel 共b兲 vacancy concentrations estimated using Eq. 共1兲 with a 2.6 mK/W resistivity factor.

whose resistivity factor is very small 共see above兲, one ends up considering that the additional thermal resistance stems mainly from the monovacancies. The ratio between carbon and silicon vacancies is not known. Assuming equal numbers of silicon and carbon vacancies, one gets an average resistivity factor of 2.6 mK/W for defects in the samples. The irradiation experiments reported from the literature in Fig. 3共a兲 were performed at various doses and various temperatures that were always higher than the temperature at which the conductivity was measured 共room temperature兲. Therefore, the defect concentration in the thermal conductivity measurements is frozen to what it was in the irradiation experiments. Considering a resistivity factor of 2.6 mK/W, Eq. 共1兲 provides us with a direct relationship between the measured thermal conductivities and the defect concentrations in the samples. This finally leads us to a scale for the concentration of defects indicated in Fig. 3共b兲. Note that this scale provides an upper bound estimate for the vacancy concentrations in view of the above mentioned assumptions. One can see that the concentration of vacancies deduced from this scales varies by two orders of magnitude depending on irradiation conditions. It varies from 10−3 for low doses 共about 5 ⫻ 1024 neutrons/ m2兲 and high irradiation temperatures 共800 ° C兲 to 0.04 for saturation doses 共higher than 1026 neutrons/ m2兲 and irradiation temperature 共200 ° C兲 just above the critical temperature for amorphization. Point defects concentration increases under the dose accumulation up to a saturation value which depends on irra-

diation temperature. Irradiation damage at high temperatures saturates at lower defect concentrations 共about 0.02兲 than low temperature irradiations. This is due to the dynamical annealing of damage, which is all the more active at high temperature so that the balance between defect creation by irradiation and damage recovery settles at a lower defect concentration for higher temperatures. Our results are consistent with swelling experiments in irradiated SiC. Under equivalent assumptions as above, one can reasonably consider that swelling is directly proportional to the concentration of vacancies. Thus the observed linear variation in thermal resistance with swelling14 confirms the proportionality between thermal resistance and defect concentration 关Eq. 共1兲兴. Moreover, the upper bound of 4% for the concentration of vacancies obtained with our calculations is qualitatively coherent with the observed saturation of low temperature swelling at 2.5%. In conclusion, using silicon carbide as an example, our calculations indicate that the thermal resistivity due to a population of a given type of point defects in an insulating material is proportional to their concentration up to very large concentrations. The proportionality factor varies broadly with the defect type but not so much with temperature. Based on rough but reasonable assumptions about the nature of defects most prevalent under irradiation in the intermediate temperature regime, we built a scale to estimate the vacancy concentrations in irradiated SiC. This showed that the concentration of defects varies by two orders of magnitude depending on the irradiation conditions and can be as high as about 4 at. % for low temperature irradiation just above amorphization critical temperature, in qualitative agreement with swelling data. 1

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