Impact of local composition on the energetics of E-centres in Si1−xGex alloys

The energetics of the defect chemistry and processes in semiconducting alloys is both technologically and theoretically significant. This is because defects in semiconductors are critical to tune their electronic properties. These processes are less well understood in random semiconductor alloys such as silicon germanium as compared to elementary semiconductors (for example silicon). To model the random silicon germanium alloy we have employed density functional theory calculations in conjunction with the special quasirandom structures model for different compositions. Here we show that, the energetics of substitutional phosphorous-vacancy pairs (E-centres) in Si1−xGex alloys vary greatly with respect to the local Ge concentration and the composition of the alloy. The most energetically favourable E-centres have a Ge atom as a nearest neighbour, whereas the dependence of the binding energy of the E-centres with respect to alloy composition is non-linear.

correlation functions of the random substitutional alloys. Their atomistic nature ensures that there is a distribution of distinct local environments that also exist in real random alloys. For the Si 1−x Ge x alloys considered the Si or Ge atoms can be surrounded by various Si n Ge 4−n coordination shells (n ranging from 0 to 4). This forms a distribution of local environments, which in turn influence the formation and energetics of dopant-defect interactions such as the E-centre 29 . The efficacy of the SQS technique to model random alloys has been demonstrated previously in numerous systems including group IV alloys 5,20,28 , oxides 21,30 , and III-V alloys 22,31 .
The E-centre in Si or Ge is a substitutional donor atom (for example P, As, Sb) bound with a nearest neighbor vacancy. The structure and energetics of E-centres in Si or Ge have been extensively investigated using both experimental and theoretical techniques [32][33][34][35][36] . Here as a criterion for the formation of the E-centre we consider the binding energies. A defect pair such as the PV is bound if the energy difference between the pair and the isolated defects (i.e. P and V being at distances long enough that they do not associate) is negative. The more negative the binding enenrgies the more likely it would be for the PV pair to form.
Modelling E-Centres. Figure 1 is a schematic representation of the SQS cells for the Si 1−x Ge x compositions considered. These have been reported in previous work 29 , however, they have not been employed to systematically investigate the defect processes in Si 1−x Ge x alloys. Here to study the binding of the phosphorous substitutional-vacancy defect (PV pairs ot E-centres) we have calculated the energies of all the different nearest neighbour PV combinations in the seven Si 1−x Ge x alloys considered. To limit defect-image errors typically introduced in computational modelling small supercells we have constructed supercells consisting of two SQS cells each (i.e. 64 atomic sites). Therefore, the total number of PV defects considered were 896, whereas more than 1350 DFT calculations were performed to predict the binding energies. The present approach allows the thorough study of defects for a range of compositions and importantly local environment (i.e. Si-rich or Ge-rich). It is an aim of this investigation to derive trends on the impact of composition and environment around the defect on the energetics of E-centres. Figure 2 considers the effect of the nearest neighbour host atoms to the P substitutional atom. For Si 1−x Ge x cells with up to 62.5% Si content we calculate that there are also positive PV binding energies. This implies that PV pairs in Si 1−x Ge x (x ≤ 0.625) may not form in some areas of the random alloy. This is not the case for Si 1−x Ge x (x > 0.625) were the PV binding ebergy is always negative (refer to Fig. 2). This is not of course implying that it www.nature.com/scientificreports www.nature.com/scientificreports/ would be equaly likely for the PV to form at any sites in Si 1−x Ge x (x > 0.625). The more negative PV pairs will be more likely to form and more stable against dissociation. Figure 3 examines the impact of the first nearest neighbour host atoms to the lattice vacancy. The trend observed is that the E-centre will have a more negative binding energy if it forms in the vicinity of Ge atoms, irresppective of the Si 1−x Ge x random alloy composition. The physical basis of this can be traced to the lattice relaxation of the Ge host atoms (which are larger than Si atoms) in the vacant space. Figure 4 attempts to sum up the impact of nearest neighbours arounf the E-centre. This is to gain an understanding of whether the E-centres in Si 1−x Ge x prefer to form in Si-rich or Ge-rich regions of the alloy. What can be www.nature.com/scientificreports www.nature.com/scientificreports/ deduced from Fig. 4 is that E-centres form in the vicinity of one or more Ge atoms, irrespective of the alloy composition. For all the alloys considered Ge is strongly represented in the nearest neighbours around the minimum energy PV. The most energetically favourable PV defects and their nearest neighbour atoms are schematically represented (refer to Fig. 5) for every alloy considered here. The present findings are consistent with previous experimental and theoretical work where it was shown that vacancies preferentially form in the vicinity of Ge atoms 5,25,37 . Figure 6 summarizes the binding energies with respect to alloy composition. The fitted line indicates that binding energies are not linear as a function of alloy composition (i.e. Vegard's law is invalid in this case). The www.nature.com/scientificreports www.nature.com/scientificreports/ deviation from linearity of the binding energies of Si 1−x Ge x (E b SiGe ) as a function of composition can be described via the following relation: Where θ is called the bowing parameter and is calculated to be −2.0 eV. Although the trend is very similar, this bowing parameter is considerably higher as compared to the one calculated in previous work 5 . This is due to the more substantial SQS cells (32 atom SQS as compared to 16 atom SQS in ref. 5 ) and the more Si 1−x Ge x compositions used here that allow far more E-centre configurations to be considered.  www.nature.com/scientificreports www.nature.com/scientificreports/ The non-linear dependence of the binding enenrgies with respect to composition is consistent with previous work 5 , however, the present study is far more detailed given more compositions were explored, larger SQS cells and more E-centres with richer nearest neighbour environments. What is the nature of this deviation from linearity and is it determined in other defect processes in Si 1−x Ge x alloys? The Ge content dependence of the most strongly bound PV defects is in agreement with previous experimental studies for donor atom (As and Sb) diffusion in Si 1−x Ge x alloys 2,38 . In particular these experimental investigations established that the activation enthalpies of diffusion of As and Sb are not linearly dependent as a function of the Ge content of Si 1−x Ge x alloy. More recently, Kube et al 3,6 . determined the self-diffusion of Si and Ge in Si 1−x Ge x over a wide range of Ge concentration (x=0.0, 0.05, 0.25, 0.45 and 0.70) and temperatures (963 K-1543 K). These investigations 3,6 showed a non-linear dependence of the activation enthalpy of self-diffusion with Ge concentration with a bowing that is consistent with the present study. Thereafter, Saltas et al 11 . analysed the experimental results within the cBΩ thermodynamic model. In this study, Saltas et al 11 . used the thermoelastic properties (bulk modulus, mean atomic volume and thermal expansion coefficient) of Si and Ge to study the composition and temperature dependence of self-diffusion in Si 1−x Ge x . The resulting deviations from Vegard's law were attributed to the diversification of the bulk properties of Si and Ge 11 . This is anticipated to be the reason for the bowing in the binding energies of the PV defects with respect to composition calculated in the present study, however, further thermodynamic analysis will need to be performed in future work.
Apart from relaxation and thermodynamic issues the electronic properties of the defects including charge transfer can impact the properties of Si 1−x Ge x . To consider this we have performed spin polarized DFT calculations. These calculations enabled us to calculate the amount of charges on the P and its nearest neighbor atoms and plot charge densities localized on the P atoms in each configurations. The relaxed configurations of P interacting vacancies in Si 1−x Ge x are shown in Fig. 7. In each configurations, there is a substantial interaction between www.nature.com/scientificreports www.nature.com/scientificreports/ P and Si (or Ge) is observed. This is reflected in the negative Bader charge on P and positive Bader charge on Si (or Ge) (refer to Fig. 8). The amount of charge on P in each configuration is ~−3.00. In each configurations, the P forms three-coordination with adjacent Si (or Ge) atoms and they donate ~1.00 e each to P to form stable P 3− states. This is further evidenced by the charge density around the P atoms in each configurations (refer to Fig. 8). The Si-P bond distance is calculated to be ~2.30 Å while the Ge-P bond distance is ~2.40 Å. This is due to the larger atomic radius of Ge than that of Si.
summary. In the present study, electronic structure calculations were used to investigate the defect process of the E-centre in Si 1−x Ge x for a range of compositions. It is shown that the binding energies of the E-centres are strongly dependent upon the composition of the alloy, but also on the local environment around the defect. In particular, the more bound E-centres have at least one Ge atom as a nearest neighbour, whereas the dependence of the binding energy of the E-centres with respect to alloy composition is non-linear. It is shown that the more substantial 32 atom SQS can better describe the defect processes in Si 1−x Ge x as compared to 16 atom SQS cells considered previously. The present study is a paradigm of the employment of the SQS method in conjunction with systematic DFT calculations to describe non-linear energetics in random semiconductor alloys. This approach can be extended to other technologically important systems such as random alloys for materials for photovoltaics and solid oxide fuel cells.
Methods. The binding nature of phosphorous with vacancy defects in Si 1−x Ge x was examined by using the plane wave DFT code CASTEP 39,40 . The correlation and the exchange interactions are described using the www.nature.com/scientificreports www.nature.com/scientificreports/ corrected density functional of Perdew, Burke, and Ernzerhof (PBE) 41 , the generalized gradient approximation (GGA), BFGS (Broyden-Fletcher-Goldfarb-Shanno) geometry optimisation algorithm in conjunction with the ultrasoft pseudopotensials 42 . The calculations involved 64-atomic site supercell, the plane wave basis set by choosing the level of convergence of the atomic energies to 0.3 eV/atom, a 2 × 2 × 2 Monkhorst-Pack (MP) 43 k-point grid. We performed seven sets of calculations for different special quasirandom structures (SQS) configurations for the following concentrations of Si 1−x Ge x (x = 0.875, 0.750, 0.625, 0.500, 0.375, 0.250, 0.125).
To understand the electronic structures of substitutional phosphorous-vacancy pairs in Si 1−x Ge x alloys, spin polarized DFT calculations were performed using VASP code 44,45 which uses plane wave basis sets. The exchange correlation term was modeled using generalized gradient approximation (GGA) parameterized by Perdew, Burke and Ernzerhof 41 . A plane wave basis set with the cut-off energy of 500 eV and a 4 × 4 × 4 Monkhorst-Pack 43 k-point mesh which yields 36 k points. Both atom positions and simulation box were relaxed using a conjugate gradient algorithm 46 . Forces on the atoms and stress tensors in all configurations were less than 0.001 eV/Å and 0.002 GPa respectively. Semi-empirical dispersion was included in all calculation as parameterized by Grimme et al. 47 in the VASP code. Bader charge analysis 48 was used to estimate the charges on the substitutional atom and its nearest neighbours.