Theoretical investigation on the point defect formation energies in beryllium and comparison with experiments

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Theoretical investigation on the point defect formation energies in beryllium and comparison with experiments L. Ferry a, F. Virot a,∗, M. Barrachin a, Y. Ferro b, C. Pardanaud b, D. Matveev c, M. Wensing c, T. Dittmar c, M. Koppen c, C. Linsmeier c a b c

Institut de Radioprotection et de Sûreté Nucléaire PSN-RES, SAG, LETR, Saint Paul les Durance cedex 13115, France Aix-Marseille Université, CNRS, PIIM UMR 7345, 13397 Marseille, France Forschungszentrum Jülich GmbH, Institut für Energie- und Klimaforschung - Plasmaphysik, 52425 Jülich, Germany

a r t i c l e

i n f o

Article history: Received 14 July 2016 Revised 3 May 2017 Accepted 30 May 2017 Available online xxx

a b s t r a c t Beryllium will be used as a plasma-facing material for ITER and will retain radioactive tritium fuel under normal operating conditions; this poses a safety issue. Vacancies play one the key roles in the trapping of tritium. This paper presents a first-principles investigation dedicated to point defect in hcp beryllium. After showing the bulk properties calculated herein agree well with experimental data, we calculated the formation energy of a single-vacancy and henceforth propose an estimate of 0.72 eV. This value is discussed with regard to previous theoretical and experimental studies. © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction ITER [1] aims to demonstrate the feasibility of obtaining energy from nuclear fusion reactions in a magnetically confined plasma of deuterium (D) and tritium (T). The low burning efficiency of the D-T reaction implies that large amounts of hydrogen isotopes will have to be present in the plasma and consequently can be retained in the first-wall plasma-facing components (PFCs) of the tokamak. To minimize the radiological consequences of a possible confinement loss, the concentration of T in the vacuum vessel (VV) must be limited during normal operations. The administrative limit on T-inventory in the VV is currently fixed at ≈700 g. Assuming that about 1 g of T will be retained after each plasma shot, this limit will be reached after ≈700 shots, corresponding to only two months of normal operation [2]. As a result, an active control of the T in-vessel inventory is crucial for limiting the amount of T trapped in the tungsten divertor and in the beryllium tiles that cover the largest part of the wall (about 700 m2 ). According to previous laboratory experiments [3,4], baking the beryllium tiles could be an efficient way to partly remove the implanted hydrogen isotopes. As a consequence, understanding the thermally activated processes that lead to tritium desorption is of prior importance. In this way, Piechoczek et al. [5] proposed a model based on

∗

Corresponding author. E-mail address: [email protected] (F. Virot).

reaction-diffusion equations to simulate Thermal Desorption Spectroscopy (TDS) for laboratory experiments performed on Be (0 0 01) and Be (112¯ 0) single crystals pre-implanted with deuterium at low fluences of ≈3 × 1019 m−2 . In their model, deuterium diffusion using the second Fick’s law, formation of mono-vacancies and selfinterstitial Be atoms are introduced as source terms based on binary collisions simulations with the SDTrimSP code [6] and formation and decomposition of hydrogen-vacancy complexes (trapping and de-trapping of deuterium) are assumed to be thermally activated processes according to Arrhenius-type expressions. While mostly based on DFT calculations reported in [7], Piechoczek et al. had to correct some parameters (self-interstitial diffusion activation energy) and introduce a new reaction, so called self-trapping, to properly fit the TDS data. This indicates that the atomic-scale mechanisms governing the hydrogen behaviour in beryllium may not be understood today, resulting in an incomplete rate-equation model. Consequently, we have decided to focus our work on an indepth investigation of hydrogen and point defect behaviour. In this paper, we address the properties of empty vacancy, whereas the second topic will be investigated in the very next future. Vacancies have been shown to be the dominant defects versus interstitials [8]. The equilibrium concentration of vacancies V, cV (T) at temperature T, is a key property to simulate the hydrogen behaviour in Be and is related to the vacancy formation energy, f EV , through cV (T ) ∝ e− f E

V /k

BT

, which requires an accurate value of

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f EV . However, available experimental and calculated values for f EV are within a relatively broad range 0.81-1.13 eV of energy [7– 11]. The aim of this paper is to clarify these apparent discrepancies and to reduce the uncertainty on the DFT data. Section 2 is devoted to the description of the computational details of our simulations. In Section 3, a comparison of the calculated properties of beryllium with the available experimental data is provided. The calculation of the formation energy of a vacancy is presented in Section 3.4 and discussed in Section 4 in regard to the previous DFT and experimental studies.

Table 1 ˚ Calculated and experimental beryllium properties. Lenghts in A, cohesive energy in eV and bulk modulus in GPa. Source

a

c

c/a

Ecoh

B

Present work Experiment Wachowiczd Wachowicze

2.258 2.286a 2.230 2.260

3.549 3.584a 3.510 3.550

1.572 1.568 1.573 1.570

3.70 3.32b 4.20 3.74

121 121c 128 115

a b c d

2. Computational details

e

2.1. Non defective beryllium bulk The calculations in this work are based on DFT as implemented in the Quantum Espresso Package [12]. The Perdew–Burke– Ernzerhof functional [13] built in the generalized gradient approximation (GGA) is used to compute the exchange and correlation energies. The same ultra-soft pseudopotentials [14] (USPPs) as in [15] were used to model the ionic cores. Only the 2s electrons of beryllium are explicitly considered, while the 1s2 electrons were included in the pseudo-potentials. The ionization energy of the first core electrons (≈112 eV) is much higher than the one of the two 2s (9.3 eV for the first ionization and 18.2 eV for the second one [16]), which should insure a high transferability of the pseudopotential. Results related the ground state properties of bulk beryllium presented in Section 3 validate the used USPPs. All the atoms are included in the optimization procedure; they are allowed to relax until the residual force fell below 0.003 eV/A˚ and the total energy below 0.001 eV. The sampling of the Brillouin–Zone (BZ) was done using the  -centered Monkhorst–Pack k-point grid [17]. [11] showed that a high number of k-points is needed in order that the bulk properties converge well. Consequently, we used the same parameter as Ganchenkova et al. [11] for a unit cell containing 2 atoms, with Ecut =816 eV and k-point number equal to 303 . It was nevertheless possible to reduce the sampling to 243 k-points for the unit cell with no loss of accuracy, the ground state energy being the same within 1 meV/atom. The Marzari–Vanderbilt smearing scheme [18] was used with a broadening of σ = 0.001 Ry . The calculated lattice parameters as well as the equilibrium volume 0 were kept frozen in the next steps of our optimization process. Keeping the previous k-mesh sampling, we tested the effect of different Ecut values on the convergence of the ground state energy. Values ranging from 10 to 60 Ry were tested, leading to an optimum value of 30 Ry (408 eV), for which the ground state energy is close to the value obtained for Ecut =60 Ry (816 eV), within 0.15 meV/atom. 2.2. Defective beryllium bulk Vacancy properties are known to be highly sensitive to the DFT parameter used. Consequently, the consistency of these parameters was checked with regard to the formation energy of a vacancy, f E1V , defined as :

 f E 1V = E (N − 1, 1V, N0 ) −

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N−1 E (N, 0V, N0 ) N

(1)

where E (N − 1, 1V, N0 ) is the total energy of the defective supercell with N − 1 atoms and one vacancy at volume N0 , and E(N, 0V, 0 ) is the total energy of the non defective supercell with N atoms and 0 . For the calculation of f E1V , it is required that the size of the supercell is large enough to minimize the interaction of the vacancy with its own image in the neighbouring cells. Keeping the

Experimental values from:[19]. [20–22]. extrapolated at 0 K [23]. Calculated ones from: LDA [24]. GGA [24].

same grid of k-points and a smearing of 0.001 Ry as for the non defective cell, it was not possible to reach convergences on f E1V considering supercells size up to 5 × 5 × 5. The reason is a too loose sampling of k-points. We found a well-balanced compromise between sampling and smearing using a 203 k-points in conjunction with a smearing of 0.05 Ry to reach the convergence. This point is discussed Section 3.2. 3. Calculated beryllium properties 3.1. Structural properties, bulk modulus, and cohesive energy From ambient temperature up to 1530 K at 0.1 MPa, beryllium has a hexagonal close-packed (hcp) structure that belongs to the P63 /mmc space group. From Tα −β =1530 K to the melting point, Tm =1560 K, it has a body-centred cubic (bcc) structure that belongs to the Im3m space group [25]. Only the lowtemperature phase will be considered in the present work. Amonenko [26] performed a critical assessment of the available lattice parameter measurements for the hcp phase, showing the influence of impurities and uncontrolled atmosphere. He finally measured a=2.2804 A˚ and c=3.5775 A˚ under vacuum on a distilled beryllium sample. Later on, the lattice parameters of high purity beryllium were obtained at 298.15 K by means of X-ray diffraction, a = 2.2858 A˚ and c= 3.5843 A˚ [19], in agreement with the previous determination. The linear coefficients of thermal expansion of beryllium published by Gordon’s [27] over a large temperature interval allows extrapolation from the previous data to estimate the ˚ The crystal palattice parameter at 0 K, a≈2.281 A˚ and c≈3.579 A. rameter we calculated are a = 2.258 A˚ and c = 3.549 A˚ in good agreement with the extrapolated values at 0K. The calculated c/a ratio of 1.57 is close to the experimental determination and lower than the ideal value of 1.633 for hcp-type structure. As reported in Table 1, our values practically coincide with the previous calculations performed within the GGA approximation [24,28,29]. The LDA approximation [24,28] provides rather satisfactory lattice parameters for beryllium, however, in accordance with a known general tendency, they are systematically lower than the experimental values. The bulk modulus, B, has been computed by applying an uniform pressure, p, on the unit cell. The evolution of the total energy vs. the volume leads to the bulk modulus according to:

B = −0

dp d2 E = 0 , d d 2

(2)

where 0 is the volume at the equilibrium geometry. Our value (121 GPa) agrees with the extrapolated data at 0 K from recent measurements carried out by resonance ultrasound spectroscopy [23]. Our results are also in agreement with other GGA-based calculations ([30] and see therein references). The calculated cohesive energy per atom is defined as the difference in energy between an isolated beryllium atom, Eat , and

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1.6

-3

α1+β1r

1.3

0.86

1.5

(eV)

1.2

(eV)

1

1V

ΔfE

1.2

0.9

1.1

0.8 0.78

0.8

4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 distance between two vacancies (Å)

1

0.82

Δ fE

ΔfE

1V

(eV)

1.3

0.84

1.1

1V

1.4

3

0.76 0.74

0.9

0.72 0.8 0

25

50

75

100

125

150

number of atoms

175

200

225

250

Fig. 1. Monovancy formation energy as a function of the supercell size for smearing value of σ =0.05 Ry.

0

0.01

0.02

0.03

σ (Ry)

0.04

0.05

Fig. 2. Monovacancy formation energy trend in function of the smearing value σ .

that of the crystal per unit atom :

Ecoh NEat − E (N, 0V, N0 ) = N N

(3)

The obtained value, 3.70 eV, is significantly higher than the experimental one, 3.32 ± 0.07 eV recommended in [20–22]. The correction due to the zero-point vibrational energy has to be taken into account considering the relatively high Debye’s temperature of Be (D = 1453 K [23]). This correction is equal to 0.092 eV/atom in agreement with previous estimates (0.093 eV/atom in [31]). According to the Debye’s model [32], it can be estimated as 98 kB D with a correction of 0.14 eV/atom, consistent with the calculations. Consequently, the corrected Ecoh is equal to 3.61 eV/atom. 3.2. Formation energy of a monovacancy Fig. 1 provides the formation energy of a mono vacancy versus the supercell size given in the number of atoms contained within the cell. Convergence is reached for 128 atoms (4 × 4 × 4) with  f E 1V =0.866 eV. The formation energy of a vacancy is also plotted versus the distance between two vacancies d in Fig. 1. A ∝1/d3 behaviour seems to be identified. This is the result of geometric and/or electronic effects, meaning the consequence of an elastic deformation of the unit-cell or the generation of electronic oscillations around the vacancy. The previous value is the vacancy formation energy at constant volume. The effect of the volume relaxation can be evaluated by minimizing the energy of the supercell with a vacancy, leading to a volume contraction,  and a decrease in energy, Erelax [33] :

 Erelax ≈ p +

 0

2

B 0 + O (3 ) 2 (N − 1 )

(4)

For p=0, the volume contraction is equal to  =0.54 A˚ 3 . It allows the calculation of the volume of the vacancy formation, equal to  f 1V = 0 −  =7.29 A˚ 3 . For p=0, the decrease in energy, Erelax = 4 meV, is very low. It can be explained by the low values of B and 0 . The formation enthalpy of the vacancy (at constant pressure p=0) is then given by  f H 1V =  f E 1V − Erelax . Since f H1V does not really differ from f E1V and the variation of volume, , is very low, enthalpy values experimentally obtained at atmospheric pressure are comparable to our calculated energies at constant supercell volume. It must be underlined that the smearing value of σ =0.05 Ry (0.68 eV) may appear relatively high in our calculations, even this value is usually not given in previous papers. When utilizing a

Fig. 3. Configurations of considered divacancy, Vaa , Vac , and Vcc .

non-zero smearing value, the calculation minimizes a functional of the free energy F with respect to the total electronic density, not the total energy E. We are in fact interested in  f E 1V (σ = 0 ). Additional calculations with a reduced smearing parameter were carried out until σ =0.01 Ry (0.14 eV); they were performed on the (4 × 4 × 4) supercell with appropriate meshing in the BZ to ensure convergence1 . The convergence is nearly reached at 0.01 eV (Fig. 2) at  f E 1V =0.72 eV which can be assumed as an upper limit. 3.3. Divacancy formation energy To compare with experimental data, we will need the calculation of the formation energy of divacany defined as follows :

 f E 2V = E (N − 2, 2V, N0 ) −

N−2 E (N, 0V, N0 ) N

(5)

We have computed f E2V for three different configurations (aa inplane 2V, ac and cc out-of-plane 2V, Fig. 3) in a 4 × 4 × 4 supercell and a (73 ) grid of k-points with a 0.01 Ry smearing broadening. We found rather similar values, i.e. 1.83, 1.94 and 1.75 eV respectively. If dis H2V stands for the dissociative energy of the divacancy defined as dis H 2V = 2 f H 1V −  f H 2V corresponding to the reaction 2V ↔ 1V + 1V, dis H2V can be deduced from the previous values; it is −0.39, −0.50 and −0.31 eV, for aa, ac and cc, in roughly agreement with previous calculations [11]. This indicates that the formation of 2V is likely not a favourable process at low temperature. This seems qualitatively in agreement with “out-of equilibrium” observations of samples irradiated by neutrons (>1 MeV) at 77 K [34], with a limited helium concentration (<1 appm), in which vacancy clusters are not observed. A similar behaviour is described for samples irradiated by electrons (2–3 MeV) at 20 K [34]. 1 In these calculations, as for the calculation with σ =0.05 Ry, we used the same smearing value for both defective and non-defective supercells.

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L. Ferry et al. / Nuclear Materials and Energy 000 (2017) 1–5 Table 2 Vacancy formation energy in different studies. Source

XC-functionala

Energy cutoff (eV)

k-point mesh

Atom numberb

f E1V (eV)

Present work Experimentsc Krimmeld Allouche et al.e Zhang et al. f Ganchenkova et al.g Middleburgh et al.h

PBE — LDA PBE PW91 GGA PBE

408 — — 435 500 300 450

7 × — — 6 × 5 × 14 × 6 ×

128 — 36 64 96 200 200

0.72 0.75 ± 0.20 1.13 0.96 0.95 0.81 1.09

a b c d e f g h

7×7

6×6 5×5 14 × 14 6×6

Exchange-correlation energy functional. atom number is related to the non-defective supercell. see discussion. [9]. [7]. [10]. [11]. [8].

4. Discussion 4.1. Comparison with other calculations Table 2 displays the available data in the literature for the formation energy of a vacancy. Unfortunately, neither the smearing values nor the broadening scheme is given in these works. Krimmel [9] et al. obtained the largest value as can be expected from LDA. The Allouche’s value [7] is significantly higher than our; it was calculated with a supercell of 64 atoms, which is below the convergence criterion of 128 atoms for the formation energy of the vacancy. The difference with the value above 1 eV of Middleburgh et al. [8] using similar DFT parameters than Ganchenkova remains unexplained at this stage. The Zhang’s value [10] is larger than our estimate too; the exchange and correlation functional PW91 they used is very similar to PBE and should not give markedly different results [35]; using the same parameters as they did with a 0.05 Ry smearing and the PBE functional, we obtained f E1V between 0.87 for a supercell of 54 atoms and 0.94 eV for 96 atoms (Fig. 1), against 0.95 eV for Zhang with the PW91 functional. 4.2. Comparison with experimental data The comparison between the formation energy of a vacancy f E1V = 0.72 eV we calculated and the experimental value is not straightforward. It can be indirectly determined from the activation Be ), assuming this process happens acenergy of self-diffusion (Hsel f cording to a mechanism dominated by monovacancies. In that case, Be =  H 1V + H 1V , the following relation can be established: Hsel f d f where Hd1V is the activation energy for the diffusion of the vacancy. Be and H 1V are deterConsequently, f E1V can be known after Hsel d f mined, which is a very difficult task since the monovacancy contribution has to be separated from contributions of other defects. The very scarce data [34,36] on vacancy diffusion in hcp beryllium have to be cautiously considered in order to determine Hd1V . In both studies, beryllium grade (SR Pechiney, ≈0.1%), supplied by the same manufacturer, was relatively low and can significantly impact Hd1V . The value, usually cited, around 0.8 eV, was deduced from the activation energy of resistivity recovery (in the temperature range 220–300 K) in samples irradiated at low temperatures by neutrons (>1 MeV) [34,37]. This activation energy is classically linked with the activation energy of the disappearance of the sursaturation of defects, which can be of several types. Nicoud [34,38] assumed an activated-monovacancy mechanism since its Be determined otherwise [39] (see therevalue was about half of Hsel f after). According to the author himself, there is no clear evidence of a only one type-defect mechanism. A lower value, about 0.65 eV,

was proposed by Chabre [36], based on the analysis of spin relaxation by NMR (between 300 and 1200 K) and attributed to vacancy migration at the Larmor frequency. This value is preferentially retained, Hd1V = 0.65 ± 0.2 eV to be consistent with our assessment of self-diffusion data thereafter. To be in spite of these uncertainties, the experimental values match relatively well with the DFT 1V =0.72 eV and H 1V =0.89 eV) [8]. values (Hd,  d,⊥ Be The available experimental measurements of self-diffusion Hsel f in beryllium are more numerous. Some of them, obtained by means of the 7 Be radiotracer technique, [39], show that this process has a slighly anisotropic character originating from the hcp structure. The Arrhenius activation energies are derived from [40] on a monocrystal over a significant interval of temperature for a better precision (between 840 and 1320 K). They are 1.63 and 1.71 eV for perpendicular and parallel diffusions to the basal plane, respectively. Nevertheless later on, lower values were obtained by Chabre et al. on the basis of NMR experiments performed at lower temperatures (30 0–120 0 K): 1.4 ± 0.1 eV [41] and 1.35 ± 0.07 eV [36]. An explanation for the discrepancy between both experimental data could be linked to the higher purity of beryllium samples in [36] in comparison with [40] (resp. 0.01% and few 0.01%). The Be = 1.4 ± 0.1 eV. assessed value for Hsel f

An experimental value for f H1V can be finally proposed on the basis of this assessment, 0.75 ± 0.2 eV. Our calculated value, 0.72 eV, is in quantitative agreement with this experimental data. 5. Conclusion

This paper presents a DFT ab-initio investigation of hcp beryllium, mainly focussed on the accurate determination of the formation energy of monovacancy. Our best-estimated value is 0.72 eV. This value well fits the assessed experimental data and is in quantitative agreement with the previous theoretical determination from Ganchenkova et al. [11]. Furthermore, we have shown that the formation of divacancies is not energetically favorable in respect to two separated monovacancies, consistently with previous studies [11]. In a very near future, stable positions of hydrogen in non defective and defective beryllium will be re-investigated as well as the diffusion paths between these different positions. From these new data, the interpretation of the TDS laboratory experiments [5] will be considered. Acknowledgements This work is supported by Region PACA. It has also been carried out within the framework of the EUROfusion Consortium and has

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received funding from the European Union Horizon 2020 research and innovation program. The views and opinions expressed herein do not necessarily reflect those of the European Commission. The authors acknowledge the computer time grant provided by IFERCSC, Rokkasho, Japan, in the frame of the Broader Approach in the Field of Fusion Energy Research. References [1] J. Wesson, Tokamaks, third ed., Oxford University Press, 2004. [2] M. Shimada, R.A. Pitts, S. Ciattaglia, S. Carpentier, C.H. Choi, G. Dell Orco, T. Hirai, A. Kukushkin, S. Lisgo, J. Palmer, W. Shu, E. Veshchev, J. Nucl. Mater. 438 (2013) S996. [3] M.J. Baldwin, K. Schmid, R.P. Doerner, A. Wiltner, R. Seraydarian, C. Linsmeier, J. Nucl. Mater. 337 (2005) 590. [4] K. Sugiyama, J. Roth, A. Anghel, C. Porosnicu, M. Baldwin, R. Doerner, K. Krieger, C.P. Lungu, J. Nucl. Mater. 415 (2011) S731. [5] R. Piechoczek, M. Reinelt, M. Oberkofler, A. Allouche, C. Linsmeier, J. Nucl. Mater. 438 (2013) 1072. [6] W. Eckstein, R. Dohmen, A. Mutzke, R. Schneider, Technical Report, 12/3, IPP, 2007. [7] A. Allouche, M. Oberkofler, M. Reinelt, C. Linsmeier, J. Phys. Chem. C 114 (2010) 3588. [8] S.C. Middleburgh, R.W. Grimes, Acta Mater. 59 (18) (2011) 7095. [9] H. Krimmel, M. Fähnle, J. Nucl. Mater. 255 (1) (1998) 72. [10] P. Zhang, J. Zhao, B. Wen, J. Phys. 24 (2012) 095004. [11] M.G. Ganchenkova, V.A. Borodin, Phys. Rev. B 75 (5) (2007) 054108. [12] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G.L. Chiarotti, M. Cococcioni, I. Dabo, et al., J. Phys. 21 (39) (2009) 395502. [13] J. Perdew, Physica B 172 (1) (1991) 1. [14] D. Vanderbilt, Phys. Rev. B 41 (11) (1990) 7892. [15] Y. Ferro, A. Allouche, C. Linsmeier, J. Appl. Phys. 113 (2013) 213514. [16] I. Hinz, K. Koeber, I. Kreuzbichler, P. Kuhn, Gmelin Handbook of Inorganic Chemistry : Beryllium, Supplement Volume A1, eighth ed., Springer-Verlag, Berlin Heidelberg GmbH, 1986. [17] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (12) (1976) 5188. [18] N. Marzari, D. Vanderbilt, A. De Vita, M.C. Payne, Phys. Rev. Lett. 82 (16) (1999) 3296.

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Please cite this article as: L. Ferry et al., Theoretical investigation on the point defect formation energies in beryllium and comparison with experiments, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.05.012