ECS Solid State Letters, 3 (6) P69-P72 (2014) P69 2162-8742/2014/3(6)/P69/4/$31.00 © The Electrochemical Society Impact of Plane Thermal Stress near the Melt/Solid Interface on the v /G Criterion for Defect-Free Large Diameter Single Crystal Si Growth Koji Sueoka, a, z Eiji Kamiyama, a Jan Vanhellemont, b and Kozo Nakamura a a Department of Communication Engineering, Okayama Prefectural University, Soja-shi, Okayama-ken 719-1197, Japan b Department of Solid State Sciences, Ghent University, Ghent B-9000, Belgium Veryrecently, first experimental evidence was published that the compressive thermal stress near the melt/solid interface makes a growing 300 mm diameter Czochralski Si crystal more vacancy-rich. The purpose of this letter is to explain these experimental results quantitatively by determining the dependence of the formation enthalpies of the vacancy and the self-interstitial on compressive plane stress using density functional theory based calculations. It is found that compressive plane stress gives a higher stress dependence of the so-called “Voronkov criterion” compared to the isotropic stress. In most of the central region of a growing crystal, the dominant component of the thermal stress near the melt/solid interface is compressive plane stress. The calculated plane stress dependence is in excellent agreement with the published experimental values and should be taken into account in the development of pulling processes for the mass-production of 450 mm diameter defect-free Si crystals. © 2014 The Electrochemical Society. [DOI: 10.1149/2.002406ssl] All rights reserved. Manuscript submitted March 14, 2014; revised manuscript received April 4, 2014. Published April 16, 2014. By evaluating the so-called “Voronkov criterion,” 1 Nakamura et al. 2 very recently published clear experimental evidence that the compressive thermal stress near the melt/solid interface shifts the growing Czochralski Si crystal to more vacancy-rich. According to this criterion, a crystal that is pulled with the ratio (v/G) of the pulling speed v over the axial temperature gradient G at the melt/solid inter- face, larger than a critical value (v/G) crit , is vacancy (V) rich while when (v/G) is smaller than the critical value, the pulled crystal is self-interstitial (I) rich. They evaluated the boundaries of defect-free regions experimentally with changing the pulling speed v and cal- culated the temperature distributions with the global heat transfer model. 2 The “Voronkov criterion” (v/G) crit can be written as. 3  v G  crit = C eq I (T m ) D I (T m )( E + Q I ) − C eq V (T m ) D V (T m )( E + Q V ) k (T m ) 2 (C V (T m ) − C I (T m )) , with E = E I f + E V f 2 . [1] C I , C I eq and C V , C V eq are the actual and the thermal equilibrium I and V concentrations, respectively. D I and D V are the I and V diffusivities, respectively. Q I and Q V are the reduced heats of transport of I and V, respectively, defined as the heat flux per unit flux of component atom in the absence of temperature gradient. T m is the melt temper- ature and k is the Boltzmann constant. E f I and E f V are the formation energy of I and V, respectively. One of the challenges to apply Eq. 1 in practice is the choice of intrinsic point defect formation and mi- gration energies for zero stress. 4 Recent simulations at the atomic level, support the assumption that both I and V are incorporated in the crystal at the melt/solid interface with their thermal equilibrium concentrations at T m . 5 Before the recent experimental confirmation by Nakamura et al., 2 one of the authors, claimed in 2011 6 that one should take into account the impact of thermal stresses on the intrinsic point defect parameters and thus on the critical (v/G) crit . A detailed density functional theory (DFT) study followed to evaluate the pressure dependence of both the formation enthalpy (H f ) and the migration enthalpy (H m ) of the intrinsic point defects. 7–9 It was found that the pressure induced change of H m is much smaller than that of H f . By assuming that the thermal stress is internal and isotropic in the bulk of the crystal (hereinafter referred to as “isotropic”), the ab initio calculations predicted that compressive thermal stress shifts the growing Si crystal toward more vacancy-rich. Very recently, Nakamura et al. 2 reported experimental results that were in good agreement with the predictions when assuming z E-mail: sueoka@c.oka-pu.ac.jp isotropic stress. The dominant component of the thermal stress near the melt/solid interface in most of the central region of the growing crystal is, however, internal and compressive plane stress. 10 If the local strain around a point defect is assumed to isotropic, the im- pact of stress depends on the value of mean stress independent of the stress type. However, the local strain around a point defect in the ground state, for example a vacancy of D 2d symmetry with Jahn-Teller distortion, 11 is anisotropic. In this case, the impact of thermal stress will differ whether the stress is isotropic or planar. In order to evaluate the impact of thermal stress on the critical (v/G) crit more accurately, it is therefore necessary to evaluate the dependence of the formation enthalpies of V and I on compressive plane stress near the melt/solid interface. The purpose of the present study was to obtain the dependence of the critical value (v/G) crit on plane thermal stress. DFT calculations were performed within the generalized gradient approximation (GGA) for electron exchange and correlation, using the CASTEP code. 12 The wave functions were expanded with plane waves, and the ultra-soft pseudo-potential method 13 was used to reduce the number of plane waves. The cutoff energy was 340 eV. The expression proposed by Perdew et al. 14 was used for the exchange-correlation energy in the GGA. The density mixing method 15 and Broyden Fletcher Goldfarb Shanno (BFGS) geometry optimization method 16 were used to opti- mize the electronic structure and atomic configurations, respectively. A periodic boundary condition and the P1 symmetry were imposed to the 216 atoms supercells for the calculation of perfect and defect- containing Si crystals. k-point sampling was performed at 2 × 2 × 2 special points in a Monkhorst-Pack grid, 17 which was sufficient to obtain converged results for the 216 Si-atoms supercells. 18 In case of bulk-isotropic stress, the calculation method was de- scribed in Ref. 7 and 8. For the case of plane stress, we assumed internal and (110) planar-isotropic (hereinafter referred to as “plane”) stress. The reference points were the perfect Si crystals deformed with changing cell sizes of L x = L y = L and keeping L z free. The ionic co- ordinates were fully relaxed to build up a list of (total energy: E tot )- (mean plane stress: σ ave = (σ x + σ y )/3) - (cell size and volume: L x ,L y , L z and Vol = L x L y L z ) data points. Similar calculations were performed for rectangular supercells containing point defects with changing cell sizes of L x = L y = L and keeping L z free. In the present study, only neutral intrinsic point defects with lowest energy configurations, i.e. the vacancy with D 2d symmetry and Jahn- Teller distortion 11 in Fig. 1a, and the self-interstitial [110] dumbbell (D) site, 19 were considered. In case of (110) plane stress, both the [110] D-site (with the Si dumbbell contained in the (110) plane) and the [10 ¯ 1] D-site (with the Si dumbbell not contained in the (110) plane) in Fig. 1b were calculated. ) unless CC License in place (see abstract). ecsdl.org/site/terms_use address. Redistribution subject to ECS terms of use (see 163.225.219.10 Downloaded on 2014-04-17 to IP