The CiCs(SiI)n Defect in Silicon from a Density Functional Theory Perspective

Carbon constitutes a significant defect in silicon (Si) as it can interact with intrinsic point defects and affect the operation of devices. In heavily irradiated Si containing carbon the initially produced carbon interstitial–carbon substitutional (CiCs) defect can associate with self-interstitials (SiI’s) to form, in the course of irradiation, the CiCs(SiI) defect and further form larger complexes namely, CiCs(SiI)n defects, by the sequential trapping of self-interstitials defects. In the present study, we use density functional theory to clarify the structure and energetics of the CiCs(SiI)n defects. We report that the lowest energy CiCs(SiI) and CiCs(SiI)2 defects are strongly bound with −2.77 and −5.30 eV, respectively.

It is established that carbon in Si has a strong tendency to associate with self-interstitials (Si I 's). Indeed, carbon-related defects produced by the irradiation, such as C i O i , C i C s , and C i , readily trap Si I 's leading to the formation of larger complexes (for example C i O i (Si I ) n , C i C s (Si I ) n , and C i (Si I ) n ) [12,14]. This is important, as the existence of these defects restrains the formation of larger Si I clusters, such as the (113) extended defects, which have detrimental effects in microelectronic devices [26]. Additionally, reaction processes involving the formation of C i O i (Si I ) n , C i C s (Si I ) n , and C i (Si I ) n structures are taken into account when radiation damage data are modeled for the fabrication of Si-tolerant radiation detectors [27,28].
The structure of the C i C s (Si I ) n complexes with n = 2, 3, . . . is not established, and the electrical activity of the members of the family is not known. The C i C s center has been detected and reported by many experimental techniques [28]. Regarding the optical activity, at least 11 local vibration modes have been correlated with the different configurations of the defect, which exhibit metastability [28]. However, only two IR bands have been correlated with the C i C s (Si I ) complex and another two with the C i C s (Si I ) 2 complex [16]. Regarding the electrical activity, at least two levels have been correlated with the C i C s defect [28]. However, no levels have been associated with the C i C s (Si I ) n complexes, although it is not unreasonable to expect that levels will be introduced in the gap. To the best of our knowledge, there is no experimental or theoretical study proving that Si I 's passivate the C i C s defect. Finally, the C i C s defect exhibits metastability [28], although this behavior has not been detected for the C i C s (Si I ) complex or the higher order member of the family.
In the present study, we have used extensive density functional theory (DFT) calculations to study the structure and energetics of the C i C s (Si I ) n defects (where n = 0, 1, 2). Based on these calculations, we propose the structure of the C i C s (Si I ) 2 defect.

Details of Calculations
The calculations in this study were performed using the plane wave DFT code Cambridge Serial Total Energy Package (CASTEP) [29]. To account for the exchange and correlation interactions, we implemented the corrected density functional of Perdew, Burke, and Ernzerhof (PBE) [30], within the generalized gradient approximation (GGA) and with ultrasoft pseudopotentials [31]. A 250-atomic site supercell (5 × 5 × 5 supercell with 2 atoms/primitive cell) was employed under constant pressure conditions. The cut-off of the plane wave basis set was 350 eV, and a 2 × 2 × 2 Monkhorst-Pack (MP) [32] k-point grid was used. This methodology has been employed and discussed in recent studies [33][34][35].

Definitions of Binding Energies
To distinguish between the different configurations of the C i C s (Si I ) n defect and to decide which one is favourable, the criterion is the minimization of the binding energy. For example, the binding energy to form a C i C s (Si I ) 2 defect in Si is where E[C i C s (Si I ) 2 Si N ] is the energy of an N lattice site supercell (here N = 250) containing N Si atoms, a C i , a C s atom, two Si I 's, and N Si atoms; E[C i Si N ] is the energy of a supercell containing a C i and N Si atoms; E[C s Si N−1 ] is the energy of a supercell containing one C s atom and N − 1 Si atoms; E[Si I Si N ] is the energy of a supercell containing an Si I and N Si atoms; E[Si N ] is the energy of the N Si atom supercell. A binding energy effectively means that the defect cluster is stable with respect to its constituent point defect components.

Results and Discussion
Here we have applied extensive DFT calculations to find the lowest energy structures of the C i C s (Si I ) n defects (n = 1, 2, . . . ). In essence, we used a step-by-step approach, first calculating the minimum energy structure of C i C s and then adding one and finally two Si I 's. To explore all possible configurations, we considered numerous calculations (more than 3000) by positioning the constituent defects in all possible sites in the supercell.
In the present study, we worked based on the following steps: (a) Knowing the most favourable position for C i [10], we determined the binding energy and the position of the C s substitutional defect for the near and the distant positions in Si positions in the crystal. We found two equivalent positions with a binding energy of −0.90 eV (see Figure 1a). (b) The second step was to add an Si interstitial (Si I ) to the crystal with the most favourable C s defect. We performed, for this reason, 1000 calculations in a 10 × 10 × 10 grid in a 13.5 Å × 13.5 Å × 13.5 Å cube around C i and C s defects. The binding energy of the most favourable Si I is −2.77 eV (see Figure 1b). (c) The next step was to add a second Si interstitial. Following the same procedure, we performed 1000 calculations in a 10 × 10 × 10 grid in a 13.5 Å × 13.5 Å × 13.5 Å cube in order to determine the position and the binding energy of the "second in order" Si interstitial (Si I2 ) (see Figure 1c). For a better investigation, we also performed the same procedure for the second favourable Si I . Thus, we ended up with two equivalent positions with a binding energy of −5.30 eV (see Figure 1d).

Figure 1.
Relative positions of the defects with the lowest binding energy (a) of the possible C s substitutional and the C i interstitial defect, (b) of the "first in order" possible Si interstitial (Si I ) and the chosen C s substitutional defects, (c) of the "second in order" possible Si interstitial (Si I2 ) and the chosen "first in order" most favourable Si interstitial (Si I ), and (d) of the "second in order" possible Si interstitial (Si I2 ) and the chosen "first in order" second favourable Si interstitial (Si I ). The blue circle denotes the position of the "given" interstitial and the color bar represents the binding energy for each position.
In agreement with previous DFT studies ( [36,37] and references therein), the energetically favourable structure of the C i C s defect is given in Figure 2a. In this configuration, an Si (marked Si 1 in what follows) at its lattice site separates the C i and C s atoms. The angle of C i Si I C s is 121.42 • (refer to Figure 2a). The binding energy of the C i C s defect is calculated to be −0.90 eV, in agreement with previous studies [21,36,37]. Thereafter, we formed the C i C s (Si I ) by adding an Si I to the C i C s defect. In this defect, the Si I atom is 1.88 Å away from the C i atom (d 3 in Figure 2b), whereas the distances between the carbon defects and the bridging Si atom are significantly extended compared to the C i C s defect (compare distances d 1 and d 3 in Figure 2a,b, respectively). Additionally, the presence of the Si I decreases the angle of C i Si I C s to 118.03 • . This defect is bound with −2.77 eV; i.e., there is an increase in the binding energy by −1.87 eV when an Si I is associated with the C i C s defect (refer to Table 1).  Table 1. The binding energies of the C i C s (Si I ) n defects (n = 0, 1, 2) in Si.

Title BE/eV BE Difference/eV
The addition of a further Si I results in the formation of the C i C s (Si I ) 2 defect (refer to Figure 2c). The binding energy of this defect is −5.30 eV, so there is an increase of the binding energy by −2.53 eV when an Si I is associated with the C i C s (Si I ) defect (refer to Table 1). In this configuration, the first Si I atom is again in the vicinity of the C i atom, but the d 3 distance increases to 1.95 Å (refer to Figure 2c). The second Si I atom is close to the bridging Si 1 atom (d 5 distance being 2.34 Å, Figure 2c). The introduction of the second Si I atom in the vicinity of the Si 1 atom leads to the extension of the C i Si I C s angle to 131.42 • . The distance between the two Si I atoms is 2.39 Å (d 4 distance, Figure 2c).
Here, the picture is consistent with the experimental viewpoint that the C i C s and C i C s (Si I ) defects will attract Si I . In essence, when there is a supersaturation of Si I , these defects will strongly attract and immobilize the Si I . The concentration of the C i C s (Si I ) n defects will depend upon the initial carbon concentration of Si and will also require a non-equilibrium concentration of Si I defects. The latter is because of the high formation energy of Si I under equilibrium conditions; therefore, C i C s (Si I ) n defects will acquire high concentrations under irradiation or implantation where Si I 's are more abundant. The accumulation of Si I in the C i C s (Si I ) n defects will impact the properties of Si and the concentration of unbound Si I .

Conclusions
In the present study, DFT calculations were employed to calculate the binding energies and is in essence the structure of the C i C s (Si I ) n defects (n = 0, 1, 2). The structure calculated for C i C s is in excellent agreement with previous reports. Here we have calculated the lowest energy structures of the C i C s (Si I ) and C i C s (Si I ) 2 defects by employing a comprehensive technique where all possibilities in a 250-atom supercell are considered. It is calculated that the C i C s (Si I ) and C i C s (Si I ) 2 defects are strongly bound with −2.77 and −5.30 eV, respectively. This is consistent with experimental work. The electronic structure and energetics of the derived structures should be further investigated using state-of-the-art hybrid functionals. Although the more simple approach used here will yield the right trends, it is anticipated that a hybrid functional will be more appropriate for a detailed understanding of these defects. This study mainly focuses on the binding energies of the C i C s (Si I ) n defects, but it should be stressed that kinetics can play a role in the formation of these defects. Atomistic modelling work in conjunction with thermodynamic models (mass action analysis) [38] is needed to assess the relative importance of the C i C s (Si I ) n defects as compared to competing carbon-related defects such as C i O i (Si I ) n .