Effect of Strain on Interactions of {\Sigma}3{111} Silicon Grain Boundary with Oxygen Impurities from First Principles

The interaction of grain boundaries (GBs) with inherent defects and/or impurity elements in multi-crystalline silicon play a decisive role in their electrical behavior. Strain, depending on the types of GBs and defects, plays an important role in these systems. Herein, the correlation between the structural and electronic properties of {\Sigma} 3{111} Si-GB in the presence of interstitial oxygen impurities is studied from the first-principles framework, considering the global and local model of strain. It is observed that the distribution of strain along with the number of impurity atoms modifies the energetics of the material. However, the electronic properties of the considered Si-GBs are not particularly affected by the strain and by the oxygen impurities, unless a very high local distortion induces additional structural defects.


I. INTRODUCTION
Multi-crystalline silicon (mc-Si) is the most widely used elemental material on photovoltaic technology in recent times [1][2][3]. Despite the improvement of various aspects, from purification to cell manufacturing including the silicon fabrication processes, current research aims to improve photovoltaic efficiency [4]. These improvements are primarily dominated by electrical properties of mc-Si systems, which remain limited by several types of defects, particularly the interaction between Si grain boundaries(Si-GBs) and intrinsic point defects or impurity atoms, as oxygen, carbon, nitrogen, etc. [5] and many transition metals [6][7][8].
Furthermore, some Si-GBs present residual strain, as detected by infrared polariscope or Raman spectroscopy [9,10], and relation of strain with longer carrier lifetime [11] has been reported. Moreover, the distribution of intrinsic strain and its relation to electrical activity on as-grown mc-Si has been probed experimentally in recent times as well [12][13][14]. Particularly, for metal precipitates correlation between carrier lifetime, stress and precipitate size has been observed [15]. Stress not only leads to fracture and breakage during handling and processing, but through the strain produced may also influence electrical activity and hence affect the performance of furnished solar cells [11,16]. Therefore strain play a major role, whether beneficial or not depends on Si-GBs types.
As an intrinsic impurity, large concentration of oxygen is inherent in mc-Si [17]. Oxygen atoms prefer to precipitate at grain boundaries, hence, segregation of oxygen atoms can release the stress, thus lowering the strain energy of the GB [18,19]. However, the mechanisms that control the O segregation are not yet fully understood, due to the diverse and complex nature of different GBs. A simple way to deal with this difficulty is to characterize the local structure at each site around the GBs and then to observe the connection between the boundary and these local structures in order to understand how they correlate.
In this work, we use this strategy and we investigate, from first-principles, the energetics and the electronic properties of Si-GB Σ3{111} with multiple interstitials oxygen atoms both with and without strain. We consider both tensile and compressive strain. For each configuration and type of strain, we analyze the systems revealing factors that mainly influence the system properties. We compare and discuss different strain varieties that can cause oxygen atoms to interact differently with GB. Actually, understanding how to characterize the structural properties of GBs and their correlation with the electrical activity can be fundamental to understand and control the properties of a device.

II. METHODOLOGY AND GRAIN BOUNDARY STRUCTURE
Our model of the Σ3{111} Si-GB consists of two grains of Si forming an interface along the crystallographic plane {111} (coincidence site lattice). The two Si grains are misoriented with respect to one another by an angle Ω = 60 • . In Figure 1(a) we show the Σ3{111} orthorhombic super-cell (a = b = c and α = β = γ= 90 • ) composed of 96 Si atoms. The lattice parameters are a=13.30Å, b=7.68Å and c=18.81Å. A bi-crystal [7] super-cell is adopted to follow periodic boundary conditions. A very regular structure of the Σ3{111} Si-GB, i.e. bond lengths and angles are close to the Si bulk, leads to low formation energy [19].
Concerning O atoms inclusion in the cell, we have tested many possible inequivalent positions of impurities as discussed in [19]. In this paper we focus only on the lowest energy structures (LE).
The calculations were performed using density functional theory (DFT) as implemented in the plane-wave based Vienna Ab-initio Simulation Package (VASP) [20,21]. We employed the generalized gradient approximation PBE for the exchange-correlation functional and projector augmented-wave (PAW) pseudopotentials with a cut-off of 400 eV. K-points sampling within the Monkhorst Pack scheme [22] was used for integration of Brillouin-zone together with the linear tetrahedron method including Blöchl corrections [23]. In particular, we used a k-mesh of 3×3×3 to calculate energy properties of the structures and a k-mesh of 7×7×7 to calculate their density of states (DOS). For the structural optimization, we used as threshold on the forces the value of 10 −2 eV/Å per atom. All post-processing analysis were carried out using the utilities of VESTA [24]. We always obtain the O atoms at bond-centered position between two Si atoms and, in the optimized structures, all Si atoms preserve their tetrahedral coordination [18,19]. Moreover, the interstitial O atoms can change the structural parameters such as bond lengths and angles, but total energies differ only about 0.01% irrespective of different positions around GB region. Considering that, according to literature [8,18,25], local distortion plays an important role in the segregation mechanism of O atoms, we have modeled the effect of strain on our structures. Starting from the LE Σ3{111} Si-GBs along with different numbers of interstitial O atoms (n = 1, 2, 3 and 4), the strain was applied in two different ways, global and local strain, as described in the following: (1) Global Strain (GS): in this model, we updated the lattice parameters of the LE O interstitial structure in the x and y directions leaving the z direction fixed. A schematic example of applied GS is shown in Figure 1 and compressive strain, where l corresponds to the amount of change with respect to unstrained lattice parameters of LE configuration. As the elongation or compression is rigidly applied to the whole structure, corresponding changes (elongation or compression) of all bonds in the system are of about the same amount.
(2) Local Strain (LS) [19,26]: in this model, we modified some bond lengths in the interstitial LE structure, creating locally tensile and/or compressive strain in the neighboring Si-Si bonds of O atom(s). In Figure 1(b) we show schematically by black arrows that it is a local change and details will be provided in next sections. Elongation/compression or both are labeled as LS +p%, LS −p% and LS ±p% respectively, where p corresponds to the amount of change with respect to unstrained bond length of the LE configuration. In case of LS ±p%, tensile and compressive strain are applied on two different bonds.
To investigate the interaction between the O atoms and the Σ3{111} Si-GB we calculated the impurity formation energy as This quantity has to be compared with the impurity formation energy (E OB Imp ) in bulk Si, calculated as Applying GS and LS methodologies to the LE Σ3{111} Si-GB, we constructed a series of strained structures which differ by the percentage of strain and for which we calculated the impurity formation energy as, where E n O SGB is the total energy of the strained GB including n O number of O atoms.
To keep the effect of the strain during the simulation, it is important not to fully relax the strained structures, which would prefer to relax to their unstrained counterpart [26]. Therefore comparing the energy from a self-consistent calculation and the optimized structure with imposing constrain around the O atom to preserve the distortion since they differ by about 3 × 10 −3 eV, we resort to the self-consistent calculation [19].
In  In Figure 3(b) we repeated the same analysis but for the GB with 4O atoms. We obtained the same behavior as for 1O case, although changes are weaker with respect to the unstrained structure. This is consistent with the lowering of the impurity formation energy for the 4O structure is maximum ∼ 5× 10 −2 (eV) for interstitial O atoms (n = 1 and n = 4) when the strain is less than 15.0% both global or local. Only above this limit, LS starts to have a significant impact on the electronic gap. In principle, specifically for this indeed stable Si-GB and with interstitial O impurities, vacancy defect would lead to such high local distortion or coordination defect as required to modify the electronic properties [19] significantly. Moreover, considering that, as previously discussed, LS imposes more inhomogeneous strain field compared to global strain, we have analyzed the charge redistribution through the calculation of the charge density difference (CDD), i.e. the difference between the charge density of the strained system with respect to the unstrained one. In Figure 6 (a -d) CDD have been plotted for 1O with LS +5.7%, LS ±5.7%, LS ±15.0% and LS ±30.0%. Since, this plot is of similar nature for 4O atoms, we have discussed here only the case with 1O atom.
In Figure 6 CDD is locally concentrated on the strained bonds, as expected. In particular, in Moreover, global tensile strain resulted to be relatively preferable for the GB stability due to the fact that under tensile strain silicon and oxygen can mimic the structure of SiO 2 which in fact has a larger lattice parameter than Si. Considering instead the effect of local distortion [ Figure 4, 5] while small strain is essentially ineffective, strong tension above +15%, as the one eventually produced by vacancies, is more efficient than compression in changing the electronic properties of the GBs. Actually a large tension can induce dangling bonds formation and the presence of defect states in the gap. Moreover, through the analysis of the charge density and of the electron localization function we have demonstrated that while the bond elongation reduces the charge density localized on the bond, the compression increases it. The fact that, a strain below 15.0% does not directly affect the electrical activity of this particular Si-GB, thus even with oxygen precipitates, preserving the robustness of electronic properties, could be a useful understanding in modeling devices with such samples. However, the electronic properties and the effect of interstitial oxygen(s) and applied strain may vary with the different types of Si-GB geometry and this thing need to be analyzed in more detail.

V. ACKNOWLEDGEMENTS
We would like to thank the University of Modena and Reggio Emilia for the financial support (FAR dipartimentale 2020) and Centro Interdipartimentale En&Tech, as well as the CINECA HPC facility for the approved ISCRA C project SiGB-NMI (IsC86 SiGB-NMI).