A study on density functional theory of the effect of pressure on the formation and migration enthalpies of intrinsic point defects in growing single crystal Si

ECS J. Solid State Sci. Technol.

2013

Self Cite

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...

Section: Impact Of Stress On [V/ G0] Cr I Tmentioning

confidence: 94%

“…Internal stress σ i leads to a change of the self-interstitial formation E f I and migration energy E m I given by 24 …”

Section: Impact Of Stress On [V/ G0] Cr I Tmentioning

confidence: 99%

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

ECS J. Solid State Sci. Technol.

2013

Self Cite

“…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][8][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.…”

mentioning

confidence: 95%

“…1. 7,8 Figures 2a and 2b show the formation energies E f V and E f I , respectively. It was found that both E f V and E f I are almost constant for σ ave between −1 GPa (compressive) and 1 GPa (tensile).…”

mentioning

confidence: 99%

Impact of Plane Thermal Stress near the Melt/Solid Interface on the v/G Criterion for Defect-Free Large Diameter Single Crystal Si Growth

ECS Solid State Letters

et al. 2014

Very recently, 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.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 interface, 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 calculated the temperature distributions with the global heat transfer model. 2 The "Voronkov criterion" (v/G) crit can be written as.,[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 temperature 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 migration 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 ...

“…Ab initio calculations showed that increasing the thermal compressive stress shifts the dominant type of point defects towards the vacancy due to the lowering of the vacancy formation energy while increasing that of the interstitial. 3 Due to the significant increase of the thermal stress during pulling of 450 mm-diameter Si crystals, this impact on the Voronkov criterion cannot be neglected. 4 During growth of 450 mm-diameter Si crystals, the average stress levels are indeed of the order of 30 MPa and thus much higher than 10 MPa that is maximum for typical 200 mm and smaller diameter crystals.…”

mentioning

confidence: 99%

Thermal stress induced void formation during 450 mm defect free silicon crystal growth and implications for wafer inspection

Applied Physics Letters

et al. 2013