Silicon Single Crystal Growth from a Melt: On the Impact of Dopants on thev/GCriterion

Abstract

The so called v/G criterion defines a critical value 0 crit of the ratio of the pulling rate v over the thermal gradient G at the melt-solid/interface. For a ratio larger than this critical value, the crystal is vacancy-rich while for values below the critical value, the crystal is self-interstitial-rich. When the ratio is equal to the critical value, the crystal would be defect free or made of so called "perfect silicon". The basic analytical expression for the critical ratio is analyzed and it is shown that thermal stress at the melt-solid interface and doping have an important effect on the 0 crit value. Furthermore, DFT calculations suggest that near the melt-solid interface the formation energy of the intrinsic point defects is lower than in the bulk of the crystal. This would lead to thermal equilibrium concentrations of vacancies and self-interstitials at the free surface and most probably also at the melt-solid interface that are considerably different from those in the bulk. It is illustrated how the v/G criterion can be further refined by taking into account these effects.The so called "Voronkov criterion" which allows predicting if a Si single crystal pulled from a melt will be vacancy-or self-interstitialrich, is the basis for defect-free Si crystal and wafer production. In its most simple form it is written as the critical ratio 0 crit of the pulling speed v over the thermal gradient G0 at the melt-solid interface 1-3with C eq and D the intrinsic point defect thermal equilibrium concentration and diffusivity, respectively, both at the melt temperature T m . E f is the intrinsic point defect formation energy and C the actual concentration of the intrinsic point defects at the melt-solid interface. The criterion was later further generalized by including also the possible effect of vacancy and self-interstitial drift which was shown however to be of minor importance. 3 An important question is which values one should use for the thermodynamic parameters of the intrinsic point defects in Eq. 1 as a very wide range of values have been reported in literature. The values for the self-diffusion coefficient D S D Si have however converged during the last years and in the temperature range between 855 and 1388 • C, D S D Si (in cm 2 s −1 ) is accurately described by 4with f I and f V the diffusion-correlation factors for self-interstitials and vacancies, respectively. Also the uncertainty on the transport capacity C eq D of the vacancy and the self-interstitial has decreased significantly. Assuming f I = 0.73, Bracht et al. 5,6 obtained (in cm −1 s −1 ) C eq V D V = 6.2C Si exp − 4.33 eV k B T [3] C eq I D I = 43C Si exp − 4.56 eV k B T . For the diffusivity of the vacancy Watkins recently proposed best estimates (in cm 2 s −1 ) for the double positively charged vacancy V 2+ and for the uncharged vacancy V 0 diffusivities, given by 7 D V 2+ = 6.5 × 10 −5 exp − 0.32 eV k B T [4] D V 0 = 0.0012 exp − 0.45 eV k B T . * Electrochemical Society Active Member. z E-mail: jan.vanhellemont@ugent.be Combining Eqs. 3 an...

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

“…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 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.…”

“…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.…”

“…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.…”

Impact of rapid thermal processing on oxygen precipitation in heavily arsenic and antimony doped Czochralski silicon

“…35 These concentrations of dopant-vacancy pairs are 4 and 3 orders of magnitude higher than the thermal equilibrium concentration of neutral vacancies at 1250 C which is about 3.7 Â 10 13 cm À3 at 1250 C. 34 A more extended report on this change of resistivity and its recovery during low temperature anneals will be published elsewhere. Similar trapping effects due to dopant-vacancy pair formation have been reported for Si doped with high concentrations of isovalent impurities Ge 36 and Sn.…”

“…As the As concentration is approximately 20 times higher than the Sb concentration, it can be expected that the concentration of AsV pairs is proportionally higher than that of SbV pairs as can be estimated based on a simple continuum model and is also confirmed by measuring the resistivity change after RTP. 33,34 Due to the capturing of vacancies generated during the RTP treatment by dopant atoms, the resistivity of the samples indeed increases by deactivation of dopant atoms. Assuming that the free carrier decrease is due to the formation of dopant-vacancy pairs, the total vacancy concentration in the As and in the Sb doped samples can be estimated to be about 3-5 Â 10 17 cm À3 and 4 Â 10 16 cm…”