Comment on "Experimental Study of the Impact of Stress on the Point Defect Incorporation during Silicon Growth" [ECS Solid State Lett., 3, N5 (2014)]

In a recent letter,¹ further called "the Letter", Nakamura et al presented for the first time clear experimental evidence of the effect of thermal stress on [v/G]_crit and thus on intrinsic point defect formation and migration energies, as predicted earlier based on ab initio calculation.^2,3 Several problems occur however when trying to quantify the stress effect and to compare the experimental data with theoretical predictions.

A first problem is that some of the intrinsic point defect parameters that are used in the Letter (and also in other work) where extracted from single crystal growth experiments which obviously never occur in stress-free conditions and thus implicitly already contain partly the effect of stress.⁴ Further more, it is well known that a large uncertainty exists on the pre-factors of the exponential expressions describing the thermal equilibrium concentrations (Eqs. 8 and 9 of the Letter) and the diffusivities of the point defects.

A second problem is that the point defect parameters in agreement with e.g the position of the so-called stacking fault ring or of the P_v − P_i boundary in the defect-free region, also depend on the equation that is used to express the Voronkov criterion for defect-free crystal growth. In the Letter, two expressions are given (Eq. 1 and Eq. 2, respectively). Vanhellemont² used a slightly different expression and adapted the intrinsic point defect parameters in order to obtain a [v/G]_crit value close to a published experimental value. Further more -somewhat provocatively- he assumed that the formation energies of the vacancy and the self-interstitial were the same. These parameters were only chosen for demonstrating the unexpectedly large impact of stress. By no means it was claimed that these were "the" intrinsic point defect parameters. In a later paper,⁴ very different values of the intrinsic point defect parameters were indeed chosen but still in a way that they yielded a [v/G]_crit value close to the experimentally observed one for a typical thermal stress level as illustrated in Fig. 1 of that paper. The same figure illustrates that without stress, the chosen point defect parameters lead to a [v/G]_crit that is significantly larger than the reported values.

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A third problem is the calculation of the stress near the melt/solid interface. This stress depends not only on the curvature of that interface as mentioned in the Letter but also on the value that is used for the Si Young's modulus near melt temperature. In addition, the calculated stress also depends on the simulator that is used to calculate the temperature distribution near the melt/solid interface.

There are thus several reasons why [v/G]_crit and point defect formation and migration energy values obtained by different groups, differ from each other. Each crystal grower has defined his own set that is consistent with the simulation tools and material parameters he is using.

For that reason, it is proposed that improved and self-consistent point defect parameters can be extracted by fitting the equation expressing the Voronkov criterion to the observed [v/G]_crit stress dependence. Hereby the stress dependent change of formation and migration enthalpies as obtained from ab initio calculations can be used which can easily be done by multiplying D_{I, mp}, C^eq_{I, mp}, D_{V, mp} and C^eq_{V, mp}, in Eqs. 1 and 2 of the Letter by . The a and α values that are used in this Comment are listed in Table I. E^f_V and E^f_I should be replaced by H^f_V and H^f_I as given by Eqs. 4 and 5 of the Letter, respectively. In the ideal case that the used point defect parameters are correct and there is also no error on the experimentally determined [v/G]_crit values, the a values should be 1 and this value is therefor also used as starting value for the fits.

The experimentally determined relation between thermal stress and [v/G]_crit is illustrated by the data points in Fig. 1, taken from the Letter. The observed dependence of [v/G]_crit on stress is similar to that shown schematically in Fig. 1 of Vanhellemont et al.⁴ The lines show best fits for the data points in the stress range between −5 Mpa and −18 MPa with a_i as fitting parameters and using the "stress-free" point defect parameters of Nakamura in Table I of the Letter and the adapted Eq. 2. For the data sets of crystals C and D, the upper (C_u, D_u) and lower (C_l, D_l) data point sets are fitted separately.

Hereby it should be noted that, as also mentioned in the Letter, the [v/G]_crit values for stress levels between 0 and −5 MPa were obtained near the periphery of the wafers so that lateral out-diffusion of the intrinsic point defects and the shape of the melt-solid interface will have an important influence on the experimentally observed [v/G]_crit value. As this influence is not included in the simple Voronkov criterion, the data points between 0 and −5 MPa, are not included in the fits.

The a_i values corresponding with the best fits are listed in Table II together with the coefficient of determination R². Due to the limited number and the scatter of data points and the strong cross-correlation between the a_i parameters, the uncertainty on the extracted values is quite large. Despite this, the one order of magnitude smaller a₃ values in Table II are an indication that the interstitial diffusivity value at melt temperature that was initially assumed, is an overestimation.

Although the data sets for the different crystals are rather scattered, it is clear that they have a similar slope. Performing a linear fit through all data points between −5 and −18 MPa, yields a stress-free [v/G]_crit value of 0.1676 ± 0.0013 (in mm²K^{− 1}min^{− 1}) and a slope of (1.11 ± 0.11) × 10^{− 3} (in mm²K^{− 1}min^{− 1}MPa^{− 1}). Both values are useful as rule of thumb for practical applications.

The described fitting procedure allows thus to obtain self-consistent exponential pre-factors for a given pulling process. At the same time, it is also obvious from the large scatter of the experimentally determined [v/G]_crit values in Fig. 1 and of the extracted values of the fitting parameters, that much more experimental and theoretical work is needed to obtain more reliable values of the formation and migration energies of the intrinsic point defects in stress-free Si as well as of the associated pre-exponential factors.

Hereby one could for instance also use other experimental data such as the (stress-dependent) self-diffusion coefficient D_SD that can be written as an expression of C_ID_I and C_VD_V, see e.g. Vanhellemont² and references therein. This expression could be fitted to experimental D_SD values using the same stress dependent intrinsic point defect parameters as for the [v/G]_crit fits and this approach would allow to obtain more reliable values for a and possibly even for α and the stress-free formation and migration energies of both point defects.