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.

1. Introduction

For more than six decades, the rapid evolution of microelectronics has constituted Si as a prevalent material. In the past few years, however, the aim has been to replace Si with other substrates (such as germanium) in nanoelectronic applications. Nevertheless, Si will remain the mainstream material for sensors and photovoltaics [1,2,3,4,5,6,7,8,9,10]. From a fundamental solid-state physics perspective, the study of point defects and defect clusters in Si remains of interest, as these can impact its materials properties. This is also technologically motivated, as in order to optimise devices, it is necessary to control the oxygen-related defects (vacancy-oxygen interstitial pairs, VO) as well as the carbon-related defects (such as CiCs(SiI)n and CiOi(SiI)n, n = 1, 2, …) [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25].

It is established that carbon in Si has a strong tendency to associate with self-interstitials (SiI’s). Indeed, carbon-related defects produced by the irradiation, such as CiOi, CiCs, and Ci, readily trap SiI’s leading to the formation of larger complexes (for example CiOi(SiI)n, CiCs(SiI)n, and Ci(SiI)n) [12,14]. This is important, as the existence of these defects restrains the formation of larger SiI clusters, such as the (113) extended defects, which have detrimental effects in microelectronic devices [26]. Additionally, reaction processes involving the formation of CiOi(SiI)n, CiCs(SiI)n, and Ci(SiI)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 CiCs(SiI)n complexes with n = 2, 3, … is not established, and the electrical activity of the members of the family is not known. The CiCs 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 CiCs(SiI) complex and another two with the CiCs(SiI)2 complex [16]. Regarding the electrical activity, at least two levels have been correlated with the CiCs defect [28]. However, no levels have been associated with the CiCs(SiI)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 SiI’s passivate the CiCs defect. Finally, the CiCs defect exhibits metastability [28], although this behavior has not been detected for the CiCs(SiI) 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 CiCs(SiI)n defects (where n = 0, 1, 2). Based on these calculations, we propose the structure of the CiCs(SiI)2 defect.

2. Methodology

2.1. 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].

2.2. Definitions of Binding Energies

To distinguish between the different configurations of the CiCs(SiI)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 CiCs(SiI)2 defect in Si is E where E[CiCs(SiI)2SiN] is the energy of an N lattice site supercell (here N = 250) containing N Si atoms, a Ci, a Cs atom, two SiI‘s, and N Si atoms; E[CiSiN] is the energy of a supercell containing a Ci and N Si atoms; E[CsSiN−1] is the energy of a supercell containing one Cs atom and N − 1 Si atoms; E[SiISiN] is the energy of a supercell containing an SiI and N Si atoms; E[SiN] 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.

3. Results and Discussion

Here we have applied extensive DFT calculations to find the lowest energy structures of the CiCs(SiI)n defects (n = 1, 2, …). In essence, we used a step-by-step approach, first calculating the minimum energy structure of CiCs and then adding one and finally two SiI’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 Ci [10], we determined the binding energy and the position of the Cs 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 (SiI) to the crystal with the most favourable Cs 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 Ci and Cs defects. The binding energy of the most favourable SiI 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 (SiI2) (see Figure 1c). For a better investigation, we also performed the same procedure for the second favourable SiI. 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 Cs substitutional and the Ci interstitial defect, (b) of the “first in order” possible Si interstitial (SiI) and the chosen Cs substitutional defects, (c) of the “second in order” possible Si interstitial (SiI2) and the chosen “first in order” most favourable Si interstitial (SiI), and (d) of the “second in order” possible Si interstitial (SiI2) and the chosen “first in order” second favourable Si interstitial (SiI). 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 CiCs defect is given in Figure 2a. In this configuration, an Si (marked Si1 in what follows) at its lattice site separates the Ci and Cs atoms. The angle of CiSiICs is 121.42° (refer to Figure 2a). The binding energy of the CiCs defect is calculated to be −0.90 eV, in agreement with previous studies [21,36,37]. Thereafter, we formed the CiCs(SiI) by adding an SiI to the CiCs defect. In this defect, the SiI atom is 1.88 Å away from the Ci atom (d3 in Figure 2b), whereas the distances between the carbon defects and the bridging Si atom are significantly extended compared to the CiCs defect (compare distances d1 and d3 in Figure 2a,b, respectively). Additionally, the presence of the SiI decreases the angle of CiSiICs 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 SiI is associated with the CiCs defect (refer to Table 1).

Figure 2

Schematic representation of the energetically favourable (a) CiCs, (b) CiCs(SiI), and (c) CiCs(SiI)2 configurations.

Table 1

The binding energies of the CiCs(SiI)n defects (n = 0, 1, 2) in Si.

Title	BE/eV	BE Difference/eV
CiCs	−0.90	-
CiCs(SiI)	−2.77	−1.87
CiCs(SiI)2	−5.30	−2.53

The addition of a further SiI results in the formation of the CiCs(SiI)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 SiI is associated with the CiCs(SiI) defect (refer to Table 1). In this configuration, the first SiI atom is again in the vicinity of the Ci atom, but the d3 distance increases to 1.95 Å (refer to Figure 2c). The second SiI atom is close to the bridging Si1 atom (d5 distance being 2.34 Å, Figure 2c). The introduction of the second SiI atom in the vicinity of the Si1 atom leads to the extension of the CiSiICs angle to 131.42°. The distance between the two SiI atoms is 2.39 Å (d4 distance, Figure 2c).

Here, the picture is consistent with the experimental viewpoint that the CiCs and CiCs(SiI) defects will attract SiI. In essence, when there is a supersaturation of SiI, these defects will strongly attract and immobilize the SiI. The concentration of the CiCs(SiI)n defects will depend upon the initial carbon concentration of Si and will also require a non-equilibrium concentration of SiI defects. The latter is because of the high formation energy of SiI under equilibrium conditions; therefore, CiCs(SiI)n defects will acquire high concentrations under irradiation or implantation where SiI‘s are more abundant. The accumulation of SiI in the CiCs(SiI)n defects will impact the properties of Si and the concentration of unbound SiI.

4. Conclusions

In the present study, DFT calculations were employed to calculate the binding energies and is in essence the structure of the CiCs(SiI)n defects (n = 0, 1, 2). The structure calculated for CiCs is in excellent agreement with previous reports. Here we have calculated the lowest energy structures of the CiCs(SiI) and CiCs(SiI)2 defects by employing a comprehensive technique where all possibilities in a 250-atom supercell are considered. It is calculated that the CiCs(SiI) and CiCs(SiI)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 CiCs(SiI)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 CiCs(SiI)n defects as compared to competing carbon-related defects such as CiOi(SiI)n.