JOURNAL OF APPLIED PHYSICS 101, 023527 共2007兲

Molecular dynamics modeling of the thermal conductivity of irradiated SiC as a function of cascade overlap Jean-Paul Crocombette,a兲 Guillaume Dumazer, and Nguyen Quoc Hoang CEA-Saclay, DEN/DMN/SRMP, 91991 Gif-Sur-Yvette, France

Fei Gao and William J. Weber Pacific Northwest National Laboratory, P.O. Box 999, Richland, Washington 99352

共Received 1 September 2006; accepted 17 November 2006; published online 26 January 2007兲 SiC thermal conductivity is known to decrease under irradiation. To understand this effect, we study the variation of the thermal conductivity of cubic SiC with defect accumulation induced by displacement cascades. We use an empirical potential of the Tersoff type in the framework of nonequilibrium molecular dynamics. The conductivity of SiC is found to decrease with dose, in very good quantitative agreement with low temperature irradiation experiments. The results are analyzed in view of the amorphization states that are created by the cascade accumulation simulations. The calculated conductivity values at lower doses are close to the smallest measured values after high temperature irradiation, indicating that the decrease of the conductivity observed at lower doses is related to the creation of point defects. A subsequent decrease takes place upon further cascade accumulation. It is characteristic of the amorphization of the material and is experimentally observed for low temperature irradiation only. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2431397兴 I. INTRODUCTION

SiC is a major candidate material for future fusion or fission nuclear reactors, due to its many desirable attributes for high temperature applications in a neutron radiation environment. It is, for instance, contemplated for plasma facing coatings and structural components in fusion reactors1 and as inert fuel coating or matrix in high temperature fission reactors.2 However, concern exists about the radiationinduced changes in the physical properties of this material that may have a significant influence on the design of the reactors. In such manner, the high thermal conductivity, which is one of the favorable properties of SiC, exhibits a rather large degradation under neutron irradiation 共see below兲. While it is theoretically expected that the appearance of defects by irradiation in the material will lead to a decrease of its thermal conductivity,3 experiments show that the exact nature and amount of the defects created under irradiation, which controls the degradation of the thermal conductivity, varies with temperature. The modeling of the evolution of thermal conductivity in SiC therefore requires the detailed knowledge of these defects. It is well known that, at low enough temperature, the accumulation of atomic defects eventually leads to the amorphization of the material, whereas beyond a certain critical temperature, amorphization does not occur and only atomic scale defects are produced. This critical temperature varies with the SiC polytype under consideration and with the characteristics of the irradiation 共nature and energy of the incident particle and irradiation flux4兲. For ␤-SiC 共cubic兲, the a兲

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extreme measured values are 293 K 共for 200 keV electron irradiation兲5,6 and 498 K 共Ref. 6兲 共for 1.5 MeV Xe+ irradiation兲. In this paper, we are mainly concerned with the evolution of ␤-SiC thermal conductivity in the low temperature regime where accumulation of irradiation damage will ultimately lead to the amorphization of the material. We use empirical potential molecular dynamics 共MD兲 to calculate the thermal conductivity of SiC. MD simulations give very good results for the thermal conductivity of semiconducting or insulating materials7–9 关including SiC 共Ref. 10兲兴, in which heat is transported by atomic vibrations. Moreover, MD simulations have proven a very informative tool to study the atomic mechanisms of the amorphization process. In particular, single cascade simulations11,12 have shown that 80% of the created defects are isolated point defects, whereas the remaining 20% form small clusters. Beyond these single cascade simulations, Gao and Weber13 have studied the accumulation of Si 10 keV cascades in SiC. They were thus able to describe the successive steps of the transition from the crystalline to amorphous states. They found that at lower irradiation dose, point defects are dominant and clusters of defects are rare. During continued cascade overlap, the small clusters coalesce and grow to form larger amorphous domains that become important at doses higher than 0.1 displacement per atom 共dpa兲. The complete amorphization is observed at 0.28 dpa, in very good agreement with the experiments. The calculation of mechanical properties for the damage states represented by the simulation boxes showed that the elastic constants of SiC decrease with increasing amount of damage or dose,14 which is also in good agreement with experimental values. In the present work, we calculate the thermal conductivity with accumulated cascades in these same simulation

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boxes, and thus connect the observed structural evolution under irradiation with the change in conductivity. These results are compared and discussed in view of the experimental results that are presented first. Then, the technicalities of the simulation are introduced, especially the way that thermal conductivity is calculated. Finally, the results and their analysis are presented.

III. COMPUTATIONAL APPROACH A. Thermal conductivity calculation

Thermal conductivity is defined as the ratio between the temperature gradient existing in the material and the associated heat flux. This tensoral quantity reduces to a scalar for materials of cubic or higher symmetry, JQ = − ␬ ⵱ T.

II. PRIOR EXPERIMENTAL RESULTS

Many experiments are available in literature on the variation of SiC thermal conductivity upon irradiation, dating back to Price.15 Most of the experiments involved high temperature irradiation, i.e., the SiC specimens were not amorphized. In all these high temperature studies, the conductivity decreases strongly. The amount of the decrease varies with the irradiation temperature and the nature of the specimen. Basically, the residual thermal conductivity increases with increasing irradiation temperature. For instance, in Price,15 the room temperature conductivity decreases from 62 to 8 W m−1 K−1 共respectively, 21 W m−1 K−1兲 under 7.7 ⫻ 1021 n cm−2 共E ⬎ 0.18 MeV兲 at 820 K 共respectively, 1370 K兲. Comparable results can be found in Refs. 16–19. Many experimental studies are concerned with the effect of annealing on the conductivity. Partial recovery of the conductivity starts for temperatures of the order of a few hundred kelvin, and full recovery takes place between 1300 and 1500 K.18 Low temperature irradiations are much rarer. Apart from a work by Rhode,20 the only other work is the one by Snead et al.21 Rhode irradiated SiC specimens with neutrons at 333 K but with a dose of only up to 0.07 dpa, which is too low for significant amorphization to have occurred. The thermal conductivity at room temperature was reduced to about 8 W m−1 K−1, down from 140 W m−1 K−1 before irradiation. Snead et al.21 measured the variation of the thermal conductivity in neutron irradiated SiC as a function of damage for two irradiation temperatures 共333 and 573 K兲 and various grades of SiC. Differences appeared between the two irradiation temperatures. First, the specimen irradiated at low temperature eventually became amorphous, whereas those irradiated at high temperature remained crystalline. Moreover for equal amount of damage, the conductivity decreased more for low temperature irradiation. In this last case, the thermal conductivity was found to decrease very sharply at low dpa 共down to between 15 and 30 W m−1 K−1 at 0.01 dpa兲 and then to further decrease to 3.8 W m−1 K−1 at 2.6 dpa, which was 1.6% of the original conductivity of the unirradiated specimen 共see, in particular, Fig. 2 in Ref. 21兲. In another paper,22 the thermal conductivity of fully amorphized SiC at 320 K was measured to be 3.6 W m−1 K−1. In the same paper, it was shown that partial annealing takes place at temperatures equal to or higher than 420 K for the amorphized material. After each annealing step of increased temperature, it was observed that the thermal conductivity increased when cooling down from the annealing temperature back to 320 K.

共1兲

Different methods exist to calculate thermal conductivity within MD. They all rely on the atomic scale expression of the heat flux. Silicon carbide is commonly described with empirical potentials of the Tersoff-Brenner family.23,24 In such potentials, the force on an atom i involves, for each of its neighbors j, a summation over the atoms k neighboring both i and j. Then, the heat flux is given by JQ = =

1 d V dt

冉兺 冊 E ir i

i

冋

册

1 1 Eivi + 兺 兺 共⌬F j · v j兲rij + 兺 共⌬Fk · vk兲rik . 兺 V i V i j k 共2兲

In this expression V is the volume of the box, Ei, ri, and vi are the energy, position, and velocity of atom i, respectively, and ⌬F j 共respectively, ⌬Fk兲 is the force contribution on j 共respectively, k兲 coming from its interaction with i. Various parametrization of the Tersoff potential exists for SiC. Preliminary tests were performed on perfect cubic SiC to determine which one gives the more accurate thermal conductivity. We considered three different parameter sets: the original parametrization for SiC by Tersoff,25 the one by Gao and Weber26 that was used to generate the irradiated boxes 共see below兲, and the one by Porter et al.,27 which has already been used for calculations of SiC thermal conductivity.10 For these tests, we used the so called Green-Kubo28 共GK兲 formula based on the Kubo equations.29 In this framework, thermal conductivity is deduced from the integration of the autocorrelation function of the heat flux in a box at thermodynamical equilibrium, i.e., in the absence of any temperature gradient,

␬=

V 3kbT2

冕

⬁

具Jq共t兲 · Jq共0兲典dt.

共3兲

0

More details about the technical difficulties in the implementation of this method can be found in the work by Li et al.10,30 that we follow here. However, unlike Li et al., we chose not to rescale the MD results for zero point motion, the relevance of this approximate correction being not clearly established in our opinion. Calculations were performed at various temperatures on 512 atom boxes. The MD runs lasted 4 ns. The results are summarized in Fig. 1. Even with such long time integrations, the uncertainty of the results for the GK results is large 共about 30%兲. However, one clearly observes that the parametrizations by Tersoff and Gao and Weber underestimate the thermal conductivity; whereas, the parameters by Porter et al. shows much closer agreement to experiments. For these parameters, we checked that our cal-

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FIG. 1. Thermal conductivity of crystalline ␤-SiC and comparison of various parameter sets and calculation methods. Filled symbols are for the Green-Kubo method and the open symbols are for the NEMD method.

culations are in exact agreement with those of Li et al. 共apart from the differences coming the rescaling of zero point motion; see above兲. We therefore chose to use the parametrization of Porter et al. for the following thermal conductivity calculations. As it involves the computation of fluctuations, the integral appearing in the GK method converges very slowly and thus requires very heavy computation. In practice, it can only be applied to small boxes. It proved too heavy for the boxes with accumulated cascades which contain 40 000 atoms 共see below兲. Therefore, for these boxes, we switched to so-called nonequilibrium molecular dynamics 共NEMD兲 method.31 This method relies on the addition of a supplementary term to the Newtonian equation of motion. One introduces fictitious forces on the atoms proportional to a force parameter fext, 1 mai = Fi + ˜Fi − 兺 ˜Fn . N n

共4兲

In the case of a Tersoff potential, the supplementary force is

冋

册

˜F = 兺 E fext + 兺 兺 ⌬F 共r · fext兲 + 兺 ⌬F 共r · fext兲 . 共5兲 i i j ij k ik i

i

j

k

The supplementary force induces a heat flux whereas the subtraction of its average ensures that no particle current appears in the box. After some building time, a permanent heat flux settles parallel to the direction of fext. The ratio of the heat flux to the external force parameter is directly proportional to the thermal conductivity through

冋

␬ = lim lim f ext→0

t→⬁

册

具JQ共t兲典 . Tf ext

共6兲

In the limit of infinitesimal f ext, NEMD and GK should give identical results. Note that no temperature gradient appears in the box. The NEMD method should therefore not be confused with the various direct methods, where the box is heated in some part and cooled down in some other part and the established temperature gradient is measured. In practice, the simulation boxes contain 40 000 atoms 共10⫻ 10⫻ 50 unit cells兲 initially in the cubic SiC phase. The

fictitious force was set along the large direction of the parallelepiped and the force parameter f ext was set to 3 ⫻ 106 m−1. To maintain a constant temperature within the box, a thermostat by Berendsen et al.32 was applied. Comparing the results obtained for the defect free boxes with the GK and NEMD methods, one observes that NEMD leads to quite higher values of the thermal conductivity. Beyond the large numerical uncertainties that appears in both calculations, the main reason that may be called upon to explain such differences is that the NEMD method is linear only in the limit of vanishing f ext. Some nonlinear effects may have appeared here. However, in our test cases, the results are almost identical for f ext equal to 1 ⫻ 106 m−1, the relative difference between the results for the two different forces being always less than 10%. Part of the difference between GK and NEMD results may also come from the different size of the boxes 共512 atoms for GK versus 40 000 atoms for NEMD兲. Anyway, the uncertainty associated with these differences proves negligible compared to the change of conductivity upon cascade accumulation 共see below兲.

B. Cascade accumulation

The computational methods employed to simulate cascade overlap and defect accumulation at 200 K have been previously described in detail elsewhere,13 and thus, only the central principles are recalled in this paper. Periodic boundary conditions were imposed along three directions of the MD cell. A kinetic energy of 10 keV was given to an atom near the top of the MD cell to start the initial displacement cascade, which was then allowed to evolve for about 10 ps. The MD cell was further equilibrated for another 10 ps to maintain temperature control, with the atoms in the two boundary planes being coupled to a reservoir. The second and subsequent cascades were simulated with a similar procedure, but with random atoms and directions. A total of 140 displacement cascades were modeled to reach a completely amorphous state, and the accumulated dose following each cascade was given in MD dpa calculated as the average number of displacement per 10 keV cascade in 3C-SiC, which previous detailed MD experiments33,34 have shown to be about 100 displacements, multiplied by the number of cascades, and divided by the total number of atoms in the box. We considered 28 different boxes with irradiation damage ranking from 0 to 0.35 dpa. Calculations of thermal conductivity for these defect structures produced at 200 K were done for temperatures of 300, 600, 900, and 1200 K. The initial 200 K defected structures heated at temperatures of 600– 1200 K for the calculations are not fully representative of experimental observations at comparable irradiation temperatures, because the calculations lack annealing effects between cascades that take place at such temperatures.22 However, we chose to study the variation with temperature of the thermal conductivity of these cells to perform a gendanken experiment on the variation with temperature of the thermal conductivity of amorphous SiC in the absence of annealing.

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FIG. 2. Variation of SiC thermal conductivity as a function of accumulated irradiation damage for different temperatures.

IV. RESULTS

For all the boxes, the heat flux exhibits very large oscillations of a period lower than 0.1 ps. These oscillations are easily subtracted by a 1 ps window averaging. Once averaged, one observes for homogeneous boxes 共very low MD dpa or fully amorphous boxes兲 a uniform evolution of the flux, which stabilizes after some 10 ps. However, for medium damage boxes, one observes strong oscillations of a much larger period 共around 2 ps兲. We believe such oscillations to come from inhomogeneities of the atomic structure in the boxes. To sort out these oscillations, the heat flux entering formula 共6兲 was calculated as the average over the last 10 ps of a 40 ps run. The relative uncertainty of the results has been estimated by inverting the direction of the force with the same initial velocity distribution. The relative difference 共ratio of the difference between the two results and their average兲 for the thermal conductivity on the whole increases with the damage, going from around 20% at lower doses to about 40% at higher doses. The uncertainties on the results are thus quite large, especially for the highly irradiated boxes. The variation of thermal conductivity with cascade accumulation is shown in Fig. 2. There is some noise or irregularities in the results. Beyond these irregularities, one clearly observes that the thermal conductivity strongly decreases with accumulating MD dpa. One may distinguish between two regimes: A fast decrease initially 共between 0 and 0.1 dpa兲, the conductivity at 1.2⫻ 10−2 dpa 共respectively 0.1兲 lying between 55 and 110 W m−1 K−1 共respectively, 9 and 87 W m−1 K−1兲 depending on simulation temperature; following this primarily strong reduction, a second, slower, decrease that saturates for doses larger than 0.3 dpa around 5 W m−1 K−1. An unexpected feature appears for 300 K 共and in much less pronounced way for 600 K兲 at lower doses. Indeed one observes a minimum of the conductivity at 0.0125 dpa 共after five cascades兲 followed by a maximum around 0.04 dpa. This feature is reproducible beyond the uncertainties of the calculations. Regarding the variation of the conductivity with tem-

FIG. 3. Variation of short range order parameter 共right scale兲 with cascade accumulation; total volume swelling with cascade accumulation decomposed in the point defects and amorphous fraction contributions 共left scale兲.

perature, it is generally observed that the conductivity tends to decrease with temperature over the range of accumulated dose. V. DISCUSSION

Our results are in very good agreement with the experimental observations reported above. Similar to the experimental results,21 we observe a sharp decrease of the thermal conductivity at low doses, even if the decrease is less dramatic than in the experiments. Our calculated values at 0.1 dpa are within the experimental brackets. As observed experimentally, the decrease of the conductivity at this dose is already quite large compared to unirradiated SiC: the conductivity is down to about 10% of the unirradiated material conductivity The further decrease upon cascade accumulations is also consistent with the observations by Snead et al.21 The value we determine for fully amorphous SiC 共around 5 W m−1 K−1兲 is very close to that experimentally measured 共3.6 W m−1 K−1兲.22 The final relative decrease of conductivity is very large as the conductivity is reduced to about 2% of its original value, in quite good agreement with the experimental value of 1.6%. These results can be analyzed in terms of the amorphization process for SiC, as it has been presented in details by Gao and Weber.13 At low irradiation dose 共less than 0.1 dpa兲, mainly point defects and few clusters of defects are observed; thus, point defects are dominant. This is evidenced by the relative contribution of swellings coming from point defects on one hand and from amorphous clusters on the other hand 共see Fig. 3兲. Not surprisingly, the thermal conductivity obtained at these low doses, either experimentally by Snead et al.21 or in the present calculations, is of the same order of magnitude as that measured after high irradiation temperature for which only point defects are produced. These results are also consistent with the values calculated by Li et al.10 for SiC boxes containing 0.5% of various types of point defects 共vacancies, interstitials or antisites of either atomic kind兲. They found conductivity values lying between 14 and 40 W m−1 K−1 depending on temperature and the nature of the introduced defects.

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We found no satisfactory explanation of the unexpected minimum appearing at low doses and temperature. It probably relies on the types and population of the defects that appear in the boxes and on detailed knowledge of the associated resistivities, which is not within the scope of the present study. At doses higher than 0.1 dpa the contribution of the amorphous domains becomes non-negligible then dominant 共Fig. 3兲. The conductivity further decreases down to values that are never reached after high temperature irradiation. The lower values of the conductivity obtained in the calculations, as well as in the low temperature experiments,21 are therefore typical of amorphous SiC. Regarding the variation of the conductivity with temperature, our calculations indicate that, in the absence of annealing, the thermal conductivity of amorphous SiC decreases with temperature. Such a result is coherent with experimental observations. Indeed it was observed in the study on annealing of amorphous SiC by Snead and Zinkle22 that after each annealing step of increased temperature, the thermal conductivity increased when cooling down from the annealing temperature back to 320 K. In these experiments one can reasonably assume that no annealing takes place during the cooling from the annealing temperature down to room temperature. This shows that, in the absence of annealing, the conductivity of the damaged material should decrease with temperature. This feature is quite surprising, as such a behavior is opposite of that usually observed for glasses35 in which conductivity generally increases with temperature. As nothing suggests that the amorphization process is not complete for high dose, this behavior may evidence that the amorphous structure retains some memory of crystalline order. This remaining order could be associated with the short range order that is defined as

␩=

NSi–C −1 2

共7兲

共where NSi–C denotes the number of first neighbor Si–C bonds兲. ␩ does not decrease down to 0 in the fully amorphous SiC but saturates at about 0.49 after a crystalline to amorphous transition occurs13 as indicated in Fig. 3. VI. CONCLUSION

The evolution of SiC thermal conductivity with overlapping cascades has been calculated with nonequilibrium molecular dynamics. The calculated dependence of conductivity on irradiation dose is consistent with the experimental measurements of conductivity at low temperature irradiation. This good agreement strongly supports the general description of the amorphization process that occurs in the cascade accumulation calculations. At lower dose, mainly point defects are produced, and the conductivity then decreases rap-

idly to values characteristics of point defect saturated SiC. These values also correspond to the lowest conductivity values experimentally reachable after high temperature irradiation. Upon further damage accumulation the conductivity further decreases, while the amorphous pockets grow, down to values characteristic of amorphous SiC. ACKNOWLEDGMENTS

This work was partially supported by the Office of Basic Energy Sciences under Contract No. DE-AC05-76RL01830. 1

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