Stress influence on substitutional impurity segregation on dislocation loops in IV–IV semiconductors

Computational Materials Science 114 (2016) 23–32

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Stress influence on substitutional impurity segregation on dislocation loops in IV–IV semiconductors A. Portavoce a,⇑, J. Perrin Toinin b, K. Hoummada b, L. Raymond b, G. Tréglia c a

CNRS, IM2NP, Faculté des Sciences de Saint-Jérôme case 142, 13397 Marseille, France Aix-Marseille Univ., IM2NP, Faculté des Sciences de Saint-Jérôme case 142, 13397 Marseille, France c CNRS, CINAM, Campus de Luminy Case 913, 13288 Marseille, France b

a r t i c l e

i n f o

Article history: Received 23 July 2015 Received in revised form 30 November 2015 Accepted 10 December 2015 Available online 29 December 2015 Keywords: Silicon Germanium Dislocation Segregation Simulation

a b s t r a c t The influence of stress on the distribution of slow-diffusing substitutional impurities in the vicinity of a dislocation loop in Si and Ge bulk was theoretically investigated, at the atomic scale, using the Si and Ge Stillinger–Weber potentials via Monte Carlo and kinetic Monte Carlo simulations. The dislocation loop was modeled by an extra atomic plane introduced between two (1 1 1) planes. The calculations were performed at high temperature, for which impurity diffusion was enabled. The influence of atomic size effect on Cottrell atmosphere formation was investigated considering the difference of atomic volume between Si and Ge. The dislocation loop elastic field was found to prevent the accumulation of substitutional atoms in the vicinity of the dislocation. However, the calculations suggest that substitutional impurities can occupy interstitial sites close to the dislocation loop. In this case, the elastic field surrounding the dislocation loop can promote Cottrell atmosphere formation mainly if the impurity exhibits a larger atomic radius than the matrix atoms (Si or Ge). Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction The understanding of point defect and extended defect interactions is of great interest in material science in order to improve material properties, such as plasticity in metallurgy technology [1–3], and to prevent material degradation during fabrication processes [4–6] and operations, such as when subjected to irradiations in nuclear energy technology [7–9]. For example, in microelectronics, the understanding of dislocation loops and Si self-interstitial interactions was shown to be crucial to understand dopant diffusion in Si after dopant implantation [10,11], and thus, for dopant distribution engineering [12]. In addition, atom probe tomography (APT) was used to experimentally observe, at the atomic scale, the distribution of different types of impurities such as B, As, P, and Ni in the vicinity of dislocation loops in Si [13–16]. These observations are very important since they provide information as to the effect of dislocation loops on impurity clustering and impurity-Si phase nucleation, which have direct impacts on semiconductor electrical properties. In microelectronic technology, dislocation loops are generally formed in the semiconductor after dopant implantation and activation annealing [17]. Depending on the implantation and annealing conditions, these dislocation loops can be located in (or close to) the doped ⇑ Corresponding author. E-mail address: [email protected] (A. Portavoce). http://dx.doi.org/10.1016/j.commatsci.2015.12.016 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.

region [18]. Surprisingly, APT observations showed similar behaviors for interstitial impurities (Ni atoms) diffusing at room temperature (T), and for substitutional impurities (B, P and As) diffusing at T > 750 °C. The impurities mainly accumulate at the border of the dislocation loop. In a previous work [19], the Si Stillinger–Weber potential (SW) and Monte Carlo simulations were used in order to investigate the stress distribution surrounding a dislocation loop in the Si lattice on substitutional sites and on interstitial sites, and to simulate the behavior of fast-diffusing interstitial impurities such as Ni. It was shown that in the vicinity of the dislocation loop, the majority of substitutional sites are under compression and preventing atomic accumulation, while interstitial sites located at the border of the dislocation loop exhibit lower pressures and promote atomic accumulation at the edges of the dislocation loop. Thus, fast-diffusing interstitial impurities such as Ni and Cu are expected to accumulate at the edge of Si dislocation loops forming Cottrell atmospheres, as observed experimentally [16]. However, in the case of substitutional impurities such as B and As diffusing at temperatures allowing for precipitation and phase formation in Si, Cottrell atmospheres are not expected to form on Si dislocation loops. Consequently, in the case of substitutional impurities, the experimental observations [13–15] may be explained by either impurity site exchange in the vicinity of the dislocation loop, impurities initially on substitutional sites far from the dislocation occupying interstitial sites close to the dislocation, or by a regular segregation controlled by the difference

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between Si and impurity dangling bond energies, prevailing on elastic energy [20–22]. The calculations also showed that, due to the long-range Coulomb interactions, impurity accumulation on dislocations can occur only if the impurities do not carry the same charge [19]. Thus, fast-diffusing species should not be in their usual ionized states (on interstitial sites [23]) and dopants should not be in their usual activation states (on substitutional sites [24]) when segregating on the dislocation loop. The goal of the present work is to better understand the behavior at the atomic scale of substitutional impurities in the vicinity of Si dislocation loops. In particular, the simulations aim to investigate (i) whether substitutional impurities can segregate on dislocation loop edges or migrate to interstitial sites in the vicinity of dislocation loops to form Cottrell atmospheres due to elastic energy minimization; and (ii) whether there is a behavior difference between impurities exhibiting a small atomic radius (r) and those exhibiting a large atomic radius compared to the atomic radius of the matrix atoms (size effect). With this aim, it was chosen to study the diffusion of Ge solute atoms in the vicinity of an Si dislocation loop and of Si solute atoms in the vicinity of a Ge dislocation loop. The Si–Ge system corresponds to an ideal solid solution (no compound formation), Si and Ge atoms occupying the substitutional sites of a same diamond lattice over the entire domain of mixing composition. The charge effects can be neglected as Si and Ge atoms have the same valence electron number. Consequently, the energy variations from one atomic site to another are expected to be mainly related to elastic energy variations. In addition, the diffusion kinetics of Si in Ge and Ge in Si are comparable to the diffusion kinetics of substitutional impurities dissolved in Si or Ge. Finally, the SW potential parametrizations for Si and Ge have been investigated in numerous works, allowing for the use of well-known Si and Ge SW potentials that have already been proven reliable in previous calculations. Thus, the Ge–Si system should allow the stress contribution to be investigated independently of the chemical effect also involved in case of dopant segregation on Si (or Ge) dislocation loops. For example, in the case of tetrahedral covalent chemical bonds, B atoms can be considered as ‘small’ atoms in Si (rSi
0.117 nm) with rB
0.088 nm and Dr = rB  rSi = 0.029 nm, whereas As atoms can be considered as ‘large’ atoms in Si with rAs
0.118 nm[24] and Dr = +0.001 nm. In addition, the chemical interactions between Si and B or As can be estimated in a simple manner by means of the parameter VAB = ½(eAA + eBB  2eAB), which gives the tendency and the strength of phase separation (VAB < 0) and compound formation (VAB > 0) between the A and B elements, with eAA, eBB, and eAB being the A–A, B–B, and A–B atomic pair energy, respectively [19,25,26]. Considering the simple model of a random A–B binary solid solution on a diamond lattice, and the experimental formation energy of the compounds SiB6 (
1.254 eV at1 [27]) and SiAs (
0.056 eV at1 [28]), one can deduce VSiB
+5.12 eV at1 and VSiAs
+0.11 eV at1. For the Si–Ge system, Dr = +0.005 nm in the case of Ge atoms (rGe
0.122 nm [24]) diluted in Si and Dr = 0.005 nm in the case of Si atoms diluted in Ge, with VSiGe = 0. Thus, the stress effect investigated in the Si–Ge system is expected to be overestimated compared to both cases, As and B atoms in Si. This is particularly true for B atoms, since the size difference between B and Si atoms is significantly larger than between Si and Ge atoms (
6 times), while the chemical interactions between B and Si are particularly high (
50 times the Si–As interactions). As for a previous work [19], the dislocation loop (DL) was modeled as an additional plane of finite size inserted between two (1 1 1) planes in either the Si or the Ge lattice. The SW potential was used to relax the Si or the Ge lattice with the DL, using the Monte Carlo (MC) technique. After relaxation, the same potentials were used to calculate the energy and the hydrostatic pressure for each atomic site in the crystals. Finally, the calculated energies

were used to simulate, by kinetic Monte Carlo (KMC), the diffusion of Ge atoms in the Si lattice and of Si atoms in the Ge lattice, allowing for atomic jumps on substitutional and interstitial sites, using the relaxed lattice containing the DL as a rigid lattice. Compared to the previous calculations that aimed to investigate interstitial impurity accumulation on Si DL [19], in this work: (i) the DL size was increased from 3 nm (61 atoms) to 5 nm (183 atoms) leading to an increase of the simulation cell from 16,189 to 32,768 atoms, (ii) in addition to substitutional (S) and tetrahedral interstitial (TI) sites, the hexagonal interstitial (HI) sites were also considered in the Si and Ge lattices, (iii) the diffusion of substitutional impurities occurring at temperatures for which T/Tm
0.7 (with Tm being the melting temperature of Si or Ge) [29], lattice relaxation and impurity diffusion were performed at T = 1173.15 K for Si and T = 873.15 K for Ge, instead of 10 K in the previous work [19], and (iv) all the atomic interactions were calculated using the Si–Si, Ge–Ge, and Si–Ge SW potentials using the cut-off distance corresponding to the specified potential parametrization, instead of considering first-neighbor atomic pair energies in the case of metallic atom and Si atom interactions in the previous study. We show that: (i) at high temperature, the pressure distribution on atomic sites surrounding the DL is less uniform, and the range of the DL stress effect on substitutional and interstitial atomic sites is reduced (two to three atomic planes), decreasing the segregation tendency of impurities on the DL, (ii) single S impurities can reach the DL but cannot accumulate on the DL due to high compression stress, (iii) if the impurities can change from S to TI sites, they can accumulate on both the DL plane and the DL edges, (iv) the DL stress effect is more pronounced for atoms having an atomic radius larger than that of the matrix atoms, the accumulation tendency on the DL of impurities occupying TI sites being significantly reduced in the case of ‘small’ impurities, and (v) despite the well-known overestimation of interstitial atom energy by the SW potential [26], the present calculations show that S impurities can occupy TI sites located at the border of a DL. These results suggest that the experimental observations of As accumulation at Si DLs edges [13,14] could be explained by the formation of Cottrell atmospheres resulting from the change of As atomic site occupancy from substitutional sites (far from the DLs) to interstitial sites in the vicinity of the DLs. However, atom probe tomography observations of B accumulation at Si DL edges [15] appear to be more difficult to explain only by the stress effect, even if B atoms occupy interstitial sites close to the DLs, since ‘small’ interstitial atoms are expected to be significantly less influenced by the elastic field surrounding Si DLs at high temperatures. Due to the strong Si–B chemical interactions, the reason for B accumulation on Si DL edges could be more easily explained by a regular segregation due to B–Si bonding and/or due to charge effects (Coulomb interactions) between B atoms (acceptors in Si substitutional sites) and the DL, not taken into consideration in the present calculations. 2. Computational procedure 2.1. Si and Ge Stillinger–Weber potentials The SW potential has been extensively used to model Si, Ge, and SiGe properties at the atomic scale. However, depending on the goal of the calculations, different parametrizations have been proposed [31–40]. In our case, vacancies were not considered since firstneighbor atomic exchanges were performed on a fully occupied and rigid lattice. Thus, as the simulations focused on modeling substitutional and interstitial atom interactions with a DL created as an interstitial plan inserted between two (1 1 1) planes [10,17], the choice of the SW parameters was mainly based on the properties of the simulated interstitial point defects. The formation energies of interstitial defects were calculated using Eq. (1):

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Ef ¼ ENþ1  ðN þ 1ÞEcoh ;

ð1Þ

with Ef the defect formation energy, EN+1 the energy of a relaxed simulation cell containing N atoms on lattice sites as well as a single interstitial atom (N = 512 in our case), and Ecoh the atomic cohesive energy. Table 1 presents the cohesive energy as well as self-TI (ETI) and self-HI (EHI) formation energies calculated at 10 K by MC in pure Si, using three different parametrizations proposed by Balamane et al. [35], Wang and Clancy [39], and Jian et al. [34], respectively. The values in parentheses correspond to the values calculated by the same authors at 0 K by molecular dynamic. The energy differences between point defects are also given, since energy differences between atomic sites are actually more important than quantitative energies for KMC simulations. Experimentally, for pure Si, Ecoh = 4.63 eV [24] and ETI
4.435 eV [29]. Ab-initio calculations predict 3.18 6 ETI 6 7.21 eV, with 3.36 6 ETI 6 7.21 eV in the case of uncharged TI, 4.04 6 ETI 6 6.59 eV in the case of TI with a single positive charge, and 3.18 6 ETI 6 6.0 eV in the case of TI with a double positive charge [29]. Thus, the average value of TI formation energy can be considered to be ETI
4.66 eV from the compilation of ab-initio data [29]. The SW potential is known to overestimate interstitial point defect formation energies. However, considering experiments and the significant variations of ETI obtained with the different ab-initio methods, the TI formation energy obtained using the SW potential appears to be acceptable. However, this is not the case for the value of EHI. Indeed, ab-initio methods predict that for uncharged defects EHI is about 1 eV smaller than ETI [29,30], in contrast with SW potential predictions (Table 1). Thus, the Si–Si SW potential proposed by Wang and Clancy was chosen for our calculations, since ETI is close to experimental and ab-initio results, with a smaller value for EHI, keeping the difference between ETI and EHI as small as possible (
1 eV) compared to the parametrization of Balamane et al. The parametrization proposed by Jian et al. could not be considered for our calculations, since it favors interstitial site formation over substitutional sites. Ecoh is found to be 4.18 eV at 900 °C with the Wang and Clancy parametrization. Table 2 presents similar calculations performed at 10 K by MC using nine different parametrizations of the SW potential for pure Ge, proposed by Wang and Clancy [39], Yu and Clancy [36], Jian et al. [34], Ding and Andersen [32], Wang and Stroud [33], Posselt and Gabriel [40], Yu et al. labeled as type A in Ref. [38], Yu et al. labeled as type B in Ref. [38], and Laradji et al. [37], respectively. Experimentally, for pure Ge, Ecoh = 3.85 eV[24] and Ecoh(Si)  Ecoh(Ge)
0.78 eV. Ab-initio calculations [41] show that in Ge: (i) ETI
Ecoh; (ii) ETI < EHI with ETI  EHI
0.14 eV, and thus, in a first approximation ETI
EHI
4.0 eV, and (iii) the split [1 1 0] self-interstitial state is the most stable state with a formation energy Esplit(Ge) > Esplit(Si), with Esplit(Ge)  Esplit(Si)
0.06 eV, suggesting that in a first approximation for the most stable interstitial state Esplit(Ge)
Esplit(Si)
3.5 eV. Table 2 shows that Table 1 Si cohesive energy, and TI and HI self-interstitial energies in Si calculated with different SW potential parametrizations at T = 10 K. SW potential

Ecoh

DE1 = |EHI|  |ETI| Balamane et al. 4.63 (4.63a) Wang and 4.34 Clancy (4.34c) Jian et al. 4.34 a b c

Ref. [35]. Ref. [30]. Ref. [39].

TI

HI

DE 1

4.84 (5.29b) (5.25a) 4.97 (4.98c)

6.34 (6.83b) (6.95a) 6.05

1.50

Table 2 Ge cohesive energy, and energy of TI and HI self-interstitials in Ge calculated with different SW potential parametrizations at T = 10 K. SW potential

DE1 = |EHI|  |ETI|, Wang and Clancy Yu and Clancy Jian et al. Ding and Andersen Wang and Stroud Posselt and Gabriel Yu et al., type A Yu et al., type B Laradji et al. a

Ecoh

TI

Ge DE2 = |ESi coh|  |Ecoh| 3.55 6.86

HI

DE 1

DE2

7.45

0.59

0.79

3.56 Unstable 3.77

7.16

7.37

0.21

0.78

13.82

13.61

0.21

0.57

3.75

10.09

7.83

2.26

0.59

3.77 (3.86a) 3.76 3.33 3.78

7.08 (3.98a) 6.56 6.97 13.81

7.50 (not stablea) 6.97 7.25 13.06

0.42

0.57

0.41 0.28 0.75

0.58 1.01 0.56

Ref. [40].

the formation energies of TI and HI are significantly overestimated with the SW parametrizations proposed in the literature. However, the SW Ge–Ge potential called ‘‘type A” by Yu et al. [38] was chosen here, since it proposes the lowest values for ETI and EHI, with the correct sign, and a small value for the energy difference ETI  EHI, in agreement with ab-initio results. In addition, comparing the Si and Ge results, the so-called type A Ge potential ensures that for the most stable interstitial position in the simulation (that is TI) ETI(Ge) > ETI(Si), with the smallest possible value for the energy difference ETI(Ge)  ETI(Si), in agreement with ab-initio results in the case of split self-interstitials. Ecoh is found to be 3.64 eV at 600 °C with the chosen Yu et al. type A parametrization. The potential proposed by Jian et al. [34] was found to not stabilize the diamond structure, promoting void formation. Finally, the classical averaging method was used to determine the parameters of the Si–Ge SW potential from the corresponding parameters of the chosen Si–Si and Ge–Ge SW potentials [42]. Table 3 presents the formation energies of Si–Ge point defects calculated by MC at 10 K using the chosen Si–Ge SW potential. GeS and SiS correspond to a single Ge atom on a Si substitutional site, and to a single Si atom on a Ge substitutional site, respectively. Following the same notation, GeTI and SiTI, as well as GeHI and SiHI, correspond to single atoms (Ge or Si) on TI and HI sites of the opposite matrix (Si or Ge), respectively. The values obtained for GeS and GeTI are similar to those obtained with other Si–Ge SW potentials [39]. It is interesting to note that despite the difference between these values and the ab-initio results [39,43], the energy difference between GeS and GeTI is similar for the SW potential (9.2 eV) and ab-initio calculations (8.77 eV). One can also note that in agreement with ab-initio calculations in SiGe [44], the formation energies of GeHI and SiHI are found to be very close to each other.

Table 3 Energy of Ge and Si point defects in Si and Ge, respectively, calculated at T = 10 K with the chosen Si and Ge SW potential parametrization, compared with other works from the literature. SW potential

GeS

Wang and Clancy for Si 3.83 Yu et al., type A for Ge SW at 0 Ka 3.58 Ab-initio at 0 Ka 5.23 Ab-initio at 0 Kb Ab-initio at 0 K in SiGec

1.08

4.35 a b c

Ref. [39]. Ref. [43]. Ref. [44].

GeTI

GeHI

SiS

SiTI

SiHI

5.37

6.38

4.40

5.61

6.30

5.22 3.54 3.54 Unstable

3.52 3.45

Unstable

3.33

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Table 3 shows that, using the SW potential, Ge substitutional and interstitial point defects in Si are found to be less stable than the corresponding Si self-interstitial defects (Table 1), whereas the opposite is true for Si point defects in Ge (Table 2). In contrast with classical elasticity, this could be related to the potential asymmetry between the repulsive energy related to the Pauli exclusion principle and the attractive energy related to atomic electron cloud and nucleus interactions, leading to different elastic effects in the case of a small atom diluted in a matrix of large atoms and vice versa, as already observed in metals in previous works [22,45]. 2.2. Dislocation loop relaxation As proposed in a previous work, the DL was modeled thanks to the insertion of self-TI atoms in the diamond lattice. These interstitial atoms formed a finite plane, exhibiting a hexagonal shape [19] located between two (1 1 1) planes of the lattice, forming a planar defect in the Si or Ge matrix as suggested by transmission electron microscopy [10,16,17] and APT experimental observations [16]. This plane was introduced in the center of a diamond lattice made of 32,768 atoms with a lattice parameter of 0.54311 nm, exhibiting 16,384 TI sites and 65,536 HI sites. The size (side  side  height) of the volume containing all the atoms was
8.69  8.69  8.69 nm3, and the diameter (£) of the dislocation loop was
5 nm. As the number of atomic sites available for impurities was limited by the use of a rigid lattice, the consideration of HI sites in addition to TI sites allowed the number of segregation sites around the DL to be potentially increased after relaxation. Despite the increase in simulation time, the size of the DL was chosen to be large enough (>3 nm) to exhibit a higher stability than {3 1 1} defects that precede DL formation via self-interstitial agglomeration in experiments [17], and thus, to be closer to the size of the DL on which impurity segregation was experimentally observed (
10 nm). The size of the simulation cell was chosen to prevent any influence of the DL on the atoms located at the edge of the simulation cell (i.e. the size of the simulation cell allows an isolated finite-size dislocation loop to be studied). Thus, atom positions were relaxed by MC simulations at T = 1173.15 K for Si and T = 873.15 K for Ge, using the Si or Ge SW interatomic interactions described in the previous section. These temperatures were chosen to match the diffusion temperatures of substitutional dopant impurities in Si and in Ge, and with the aim that the results obtained in Si and Ge would be comparable. These temperatures were chosen to correspond to T/Tm
0.7 for both the Si and Ge matrices. Periodic boundary conditions were used in all directions for all the calculations. The atoms in the simulation cell were sequentially chosen in a random order, and each selected atom was proposed a new random position with a maximum distance rmax
0.01 nm from its current site. The energy variation of the entire simulation cell that would result from this new position was calculated and used to accept or reject the atom’s new proposed location according to the Metropolis algorithm [46]. The relaxation was considered to be completed when the fluctuations of the energy of the entire simulation cell and of the averaged accepted atom displacement were considered to be sufficiently small to estimate that an equilibrium value had been reached in the MC simulation. After relaxation of the entire simulation cell containing the dislocation loop, the hydrostatic pressure (pi) was calculated for each lattice site, TI site, and HI site i [19].

increasing the calculation time, the method commonly suggested in the literature consists of using a continuous expression of the elastic field, in addition to the atomic pairwise description, especially in the vicinity of a dislocation [47]. In our case, we proposed using the SW potential to calculate interatomic interactions, including elastic interactions, in order to obtain an atomistic description of the stress distribution in the vicinity of the considered DL. For this purpose, a single impurity (i.e. Si or Ge atom) was placed on each atomic site S, TI, and HI in the relaxed simulation cell containing the DL, and the atom energy was calculated using the Si–Si, Ge–Ge, and Si–Ge SW potentials. Thus, the difference between these energies was used as the energy barrier in the KMC simulations. At the atomic scale, a usual model considers that the diffusion barrier can be expressed as the sum of two terms: (i) a kinetic term related to the diffusion mechanism allowing atoms to pass from one atomic site to another, and (ii) a thermodynamic term related to the change of the system’s total energy after the atomic jump. The kinetic term allows a more quantitative time scale to be defined, especially if this term varies with the type of atomic jump. However, in our case, impurities were considered to be isolated in the Ge matrix. Thus, the kinetic term could be considered as constant, and was not calculated as we were mainly interested in the atomic distribution in the DL vicinity. Absolute energy values are not essential for KMC simulation, but rather the energy difference between the case of the considered atom sitting on the initial site before the atomic jump, and the case of the same atom sitting on the final site after the atomic jump [46]. During the diffusion simulations, impurities were randomly inserted on the S sites of the simulation cell. The atoms were able to exchange with their S first-neighbor atoms on the matrix lattice, as well as to jump to their vacant TI and HI first-neighbor sites, following the Metropolis algorithm [46]. The matrix lattice was kept fully occupied by atoms, whereas the amount of impurity atoms could vary on the interstitial lattice. No vacancies were created in the substitutional lattice. After an impurity left an S site for a TI or HI site, a matrix atom was put back on the empty S site. When an impurity jumped from an interstitial position to an S position, the interstitial site was left empty, and the impurity was set to replace the matrix atom on the considered S site. Once each of the impurity atoms already in the cell had performed 106 jumps, impurities were sequentially inserted into the simulation cell. The time unit used for the simulations was the Monte Carlo cycle (MCC) as previously defined [19,25]. The diffusion simulations were performed at the same temperature as for the lattice relaxation: at T = 1173.15 K in Si and T = 873.15 K in Ge. The experimental diffusion length of a Ge atom in Si is
1.05 nm/h at T = 1173.15 K [29], and of a Si atom in Ge is
1.25 nm/ h at T = 873.15 K [49]. The migration barrier DE = Em  En was used to accept or reject atomic jumps from site n to site m, with Ei being the energy of the considered impurity on the atomic site i calculated by the SW potential. In addition to the use of a semi-empirical potential such as the SW potential, the main approximation of the present calculations is the use of energies Ei corresponding to the energy of isolated and unrelaxed impurities (rigid lattice). However, in this case, the stress effect on atom segregation could be particularly separated from chemical interactions between the diffusing atoms. Furthermore, as previously noted, only the energy differences are used in the KMC simulations, and for example, without the DL influence, the result is similar for the relaxed or unrelaxed cells, the energy difference between two sites being zero if the sites are at the same energy.

2.3. Impurity diffusion 3. Results and discussion Due to usual computational time constraints, atomic scale diffusion simulations are usually performed on a rigid lattice, using KMC simulations considering atomic pair energies, sometimes calculated using ab-initio techniques [19,25,26,47,48]. In order to take into account elastic effects on atomic jumps without significantly

3.1. Dislocation loop surrounding pressure Similar results were obtained for the relaxation of Si and Ge. For example, Fig. 1 presents the dislocation plane (DP) obtained after

A. Portavoce et al. / Computational Materials Science 114 (2016) 23–32

27

the relaxation of the Si simulation cell delimited by the dotted lines. As observed in previous calculations performed at 10 K [19], the Si TI atoms (Brown atoms) forming the DP remained close-packed (£
5 nm), exhibiting a hexagonal shape similar to experimental DLs [16]. At the considered temperatures (T/Tm
0.7), the dislocation edge does not appear to be reconstructed, showing no sign of atomic periodic structure. However, Fig. 2(a) shows that the relaxed DP (black atoms) moved from its original location, at the same distance between two (1 1 1) planes before relaxation, towards the lower (1 1 1) plane in Fig. 2(a) (gray atoms). This figure shows also an expected increase of the atomic disorder compared to the results obtained at 10 K, the diamond lattice being less ordered at higher temperature. For the chosen Si–Si, Ge–Ge, and Si–Ge SW potentials, the cut-off distances were 0.377118 nm, 0.39258 nm, and 0.384849 nm, respectively. Fig. 2(b) presents the top view of the DP and of its closest Si(1 1 1) plane, allowing lattice distortions in the vicinity of the dislocation to be observed. The interatomic distance (d) varies between 0.215 and 0.262 nm in the Si DP vicinity and between 0.210 and 0.310 nm in the Ge DP vicinity, suggesting a significant local stress field modifying the energy of the atomic sites located close to the DP. The DL formation energy was calculated using Eq. (2):

Edcl ðTÞ ¼ Etot ðTÞ  N  Ecoh ðTÞ;

ð2Þ

with Edcl the dislocation energy, Etot the simulation cell total energy, Ecoh the atomic cohesive energy at the considered temperature T, N being the total number of atoms in the cell. In Si Edcl = 315.59 eV [50], and in Ge Edcl = 421.40 eV. Considering as defective the atoms exhibiting a nearest neighbor environment significantly different from that of the bulk, namely all the DP atoms and some of their first neighbors in the nearest (1 1 1) layer (Fig. 2(b)), the DL was found to include an average of 545 defective atoms corresponding to an energy of 0.58 eV per atom [17] in Si and 0.77 eV per atom in Ge. As expected, in Si and Ge, the elastic influence of the DL is found to be reduced to a few atomic distances, and the pressure distribution around the DL is found to be less regular at high temperature. For example, the left column in Fig. 3 presents the pressure distribution around the DL on S (Fig. 3(a)), TI (Fig. 3(b))

Fig. 2. DP atoms (black spheres) in Si after relaxation at T = 1173.15 K: (a) side view with the two Si(1 1 1) planes (gray atoms) between which the DP was inserted, and (b) top view with the closest Si(1 1 1) plane.

and HI (Fig. 3(c)) sites in the case of pure Si. The cell has been sliced along the direction (0 1 1), cutting through the center of the dislocation loop, providing a side view of half of the simulation cell towards the direction (0 1 1). The stress field on lattice and interstitial sites expands over 1–3 atomic planes in the direction perpendicular to the DP at T = 1173.15 K instead of over 5 atomic planes at 10 K [19]. However, as observed at 10 K, the stress field vanishes rapidly from the dislocation edges along the DP direction. After relaxation, all the interstitial sites are under compression, as well as the majority of the lattice sites, exhibiting positive pressures, for both pure Si and pure Ge (Fig. 4, left column). The introduction of impurities in the matrix modifies the DL elastic stress field. The right column in Fig. 3 presents the pressure distribution on atomic sites in the case of Ge atoms in solution in Si. One can note a pressure increase on both lattice and interstitial sites, as well as an increase of the stress field region size (of
2 atomic planes) around the DP compared to pure Si (Fig. 3, left column). As expected, the DL has a stronger influence on Ge atoms in solution in Si than on substitutional and interstitial Si atoms due to the larger size of Ge atoms. When a Ge atom occupies a Si site (S, TI, or HI) the pressure on the site increases by at least 1 GPa. Fig. 4 presents the pressure distribution surrounding the DL for the opposite case, when the solute impurity exhibits a smaller size than the matrix atoms (i.e. Si atoms in solution in Ge). Si atoms introduced in the Ge cell allow for a significant pressure reduction on all atomic sites. For example, the maximum pressure on S sites decreases from 5.79 GPa to 4.55 GPa, and the pressure on interstitial sites is decreased at least by 1 GPa when a Ge matrix atom is replaced by a Si solute atom. In addition, the stress field extension on lattice and interstitial sites is reduced by
2 atomic planes in the direction perpendicular to the DP. A small impurity decreases the stress field surrounding the DL, in contrast with a large impurity. 3.2. Impurity distribution close to the dislocation loop

Fig. 1. 3D distribution of the Si TI atoms (brown spheres) constituting the dislocation loop after relaxation at T = 1173.15 K. The dotted lines set the limits of the simulation cell. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

During impurity diffusion KMC simulations in the DL vicinity, the solute atoms were randomly inserted on substitutional sites, and thus, could diffuse on all the atomic sites considered in the

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Fig. 3. Hydrostatic pressure distribution on atomic sites in the Si simulation cell at T = 1173.15 K in the case of pure Si (left) and in the case of Ge atoms occupying Si sites (right). Cross-section views toward the (0 1 1) direction: (a) S sites, (b) TI sites, and (c) HI sites.

simulation cell, i.e. S, TI, and HI sites. In Si as well as in Ge, all the impurities were found to diffuse only on S sites. Impurity atoms were able to reach the DP. However, no accumulation was observed, and the majority of the inserted atoms was found to stay far from the DP. For example, Fig. 5 illustrates how the distribution of Ge atoms (black) on the Si DP (brown atoms) fluctuated over time during the KMC simulations. Similar results were obtained in the case of Si diffusion in Ge. In this figure, the simulation cell already contains 100 Ge atoms, and all the Ge atoms are substitutional atoms. They can attach to both the edge and to the plane of the DL, but their number on the DP at a given time is limited. As expected, if the impurities stay on S sites, whatever the sign of the sizemismatch, they cannot accumulate on the DL due to the stress state of S sites located close to the DL. As mentioned in Section 2.1, as the energy of TI and HI atoms is significantly larger than that of S atoms, the probability that an impurity located close to the DP can jump from an S site to an interstitial site may be seriously limited in our calculations,

especially since the energy of S atoms is compared to the energy of unrelaxed TI and HI atoms (rigid lattice after DL relaxation) during diffusion. Consequently, in order to investigate impurity behavior as if they were able to jump to interstitial sites close to the DL, KMC simulations were also performed introducing the impurities in the simulation cell on interstitial sites, and limiting atomic jumps to TI and HI sites only. Fig. 6 shows the TI (black spheres) and HI (red spheres) Ge atoms located on the Si DL (brown spheres) after a simulation time of 190,000 MCC. The simulation cell contains 100 Ge atoms on interstitial sites, and at that time the dislocation is saturated, atoms not located on the DP diffusing almost randomly in the simulation cell. In the case of interstitial Ge solute atoms in Si, the DL elastic field promotes impurity accumulation at the edges as well as on the plane of the DL. However, as expected, the accumulation is less substantial at T = 1173.15 K than at 10 K (equilibrium segregation decreases with temperature [20–22]). Fig. 7 presents the results obtained for the same KMC simulations (same simulation time and number of interstitial atoms) in the case of Si impurities diffusing in the vicin-

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Fig. 4. Hydrostatic pressure distribution on atomic sites in the Ge simulation cell at T = 873.15 K in the case of pure Ge (left) and in the case of Si atoms occupying Ge sites (right). Cross-section views toward the (0 1 1) direction: (a) S sites, (b) TI sites, and (c) HI sites.

ity of the Ge DL. Few Si atoms have reached the Ge DL. However, in contrast with Ge diffusion in Si, no significant accumulation is observed. This is in agreement with the results obtained in Section 3.1., confirming that small impurities do not experience the same stress field as large impurities in the vicinity of the DL, leading to the decrease of their segregation driving force on the DL. The same results were actually reported concerning the influence of size effect on surface segregation in the case of metals [22,45]. Fig. 8a presents the number of impurities on the DL versus impurity bulk concentration in the case of diluted Ge atoms in Si bulk (circles) and Si diluted atoms in Ge bulk (squares), if impurities occupy S sites (open symbols) or interstitial sites (solid symbols). If the impurities occupy S sites, no significant differences can be noted between the behavior of Ge impurities in Si and Si impurities in Ge. As mentioned before, atoms can reach the DL but no real segregation is observed. On the contrary, interstitial impurities segregate on the DL with a number of segregated atoms increasing with bulk concentration, as expected for an ideal solution following the McLean segregation law [51,52]. The segregation of Si atoms on the Ge dislocation is clearly smaller than the segregation of Ge

atoms on the Si dislocation. In order to quantify the segregation difference between the two investigated cases, the segregation isotherms comparing the segregation concentration and bulk concentration were plotted in Fig. 8b. The segregation concentration was expressed as the ratio between the number of atoms located on the DL and the number of available sites on the two sides of the DP. The concentration of segregated impurities (CDL) versus bulk concentration (Cb) can be fitted with a linear law in agreement with the Henry segregation law CDL = kCb [51,52], which is an approximation of the McLean law valid for strong dilutions. The segregation coefficient k was found to be about two times larger in the case of Ge segregation on Si DL. Due to the limits of using the SW potential on a rigid lattice to estimate the probability of substitutional impurities jumping on interstitial sites, two types of KMC simulations were performed in order to study the possibility of impurities occupying interstitial sites instead of substitutional sites in the DL vicinity. The same KMC simulations as previously described were performed, with the addition of the relaxation of the full simulation cell in the simulation algorithm, using the same MC method as used for the DP

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Fig. 7. 3D distribution of Si interstitial atoms (black spheres for TI and red spheres for HI) accumulated on the Ge dislocation loop (brown spheres) after diffusion KMC simulations performed at T = 873.15 K for 190,000 MCC: (a) and (b) top view of the two sides of the DP, (c) side view of the DP. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Distribution of Ge atoms (black spheres) on the Si dislocation loop (brown spheres) at different simulation times (MCC) for KMC simulations performed at T = 1173.15 K. All the Ge atoms occupy S sites. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. 3D distribution of Ge interstitial atoms (black spheres for TI and red spheres for HI) accumulated on the Si dislocation loop (brown spheres) after diffusion KMC simulations performed at T = 1173.15 K for 190,000 MCC: (a) and (b) top view of the two sides of the DP, (c) side view of the DP. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. DL coverage with impurities versus impurity bulk concentration in the case of Ge diluted atoms in Si bulk (circles) and Si atoms in Ge (squares). Open symbols and solid symbols correspond to impurities occupying substitutional sites and interstitial sites, respectively.

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relaxation described in Section 2.2. The relaxation step was performed between the atomic jumps. The goal was to investigate whether the relaxation of atoms close to the DP would favor atomic exchanges between substitutional and interstitial sites. Relaxation steps from 5 to 20 MCC were investigated, providing reasonable calculation times. However, this method provided the same results as without relaxation. This was attributed to the requirement of longer relaxation steps that would actually prevent the simulation of atomic diffusion. On the other hand, the DL atoms were fully relaxed during the MC DL relaxation calculations (Section 2.2) performed before the KMC diffusion simulations. Thus the stress state of the DL atoms located at the DP edges corresponds to fully-relaxed TI atoms. In addition, the relaxation of these atoms takes into account the presence of the other atoms in the DL, in contrast with the formation energies presented in Tables 1–3 related to isolated TI and HI atoms. In consequence, in a second attempt to consider atomic scale relaxation during diffusion on

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the rigid lattice, the DL atoms located at the DP edges were removed, leaving these atomic sites at the DP periphery available to diffusing impurities. Thus the same KMC simulations, with impurity insertion on substitutional sites, were again performed with a smaller DP, since the atomic sites located at the edges of the DP were free of atoms and thus available for diffusing species. In this case, the energy of S atoms could be compared to the energy of collectively fully-relaxed IT sites located at the edges of the DP. Fig. 9 illustrates the Ge atom distribution fluctuation versus time around the Si DP during the KMC simulations. For the sake of clarity, only half of the simulation cell is shown. The Si DP corresponds to the brown atoms, while the black and red atoms correspond to the diffusing S and TI Ge atoms, respectively. As formally observed, the Ge atom majority diffuses on S sites; however, one can note that Ge atoms can occupy the relaxed TI sites located at the border of the Si DP. Defining the time between two atomic jumps as dt = 1/C with C = D/a2, with D and a being the experimental diffusion coefficient [29,49] and the first-neighbor distance respectively, the jump frequency of Ge substitutional atoms on the TI sites located at the DL border was found to be
1.40 ± 1.5  109 s1. These results show that, despite the high energy of isolated TI atoms, the change of impurity atomic sites from substitutional to interstitial close to a DL is plausible. In this case, the impurity accumulation experimentally observed could be attributed to the formation of Cottrell atmospheres, since once the impurities are on interstitial sites, the elastic field favors their accumulation on the lower pressure atomic sites located at the DL edges. However, it is important to note that this scenario appears reasonable for substitutional impurities exhibiting an atomic radius larger than that of Si, such as As for example. In the case of impurities with an atomic radius smaller than that of Si, such as B, the formation of Cottrell atmospheres around DLs is less probable, since even if the impurities can occupy the interstitial sites located in the DL vicinity, the surrounding elastic field does not strongly promote their accumulation on the DL at high temperature. 4. Conclusion MC and KMC simulations were performed to theoretically study, at the atomic scale, the possible formation of Cottrell atmospheres with substitutional impurities in the vicinity of a dislocation loop in Si or Ge, taking atomic size effect into account. The simulations show that the elastic field surrounding dislocation loops prevents the accumulation of substitutional impurities at their diffusion temperature in Si and Ge, whatever the sign of the atomic size mismatch. However, substitutional impurities may change their atomic sites and occupy interstitial sites close to the dislocation loop. In this case, the dislocation loop elastic field can promote the formation of Cottrell atmospheres, in particular if the impurities exhibit an atomic radius larger than the matrix. These results bring additional perspectives concerning the interpretations of As and B accumulation at the edges of Si dislocation loops observed experimentally by atom probe tomography. Acknowledgements The authors would like to thank M. Portavoce for proofreading of the article. This work was supported by the French National Agency for Research (ANR) through the Program ‘‘Science de l’ingénierie” (Project DoGeTec, No. ANR-12-JS09-0015-1).

Fig. 9. 3D Distribution of Ge atoms (black spheres on S sites and red spheres on TI sites) surrounding the Si dislocation loop (brown spheres) at different simulation times for KMC simulations performed at T = 1173.15 K. Only half of the simulation cell is presented. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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