A study on density functional theory of the effect of pressure
on the formation and activation enthalpies
of intrinsic point defects in growing single crystal Si
Koji Sueoka, Eiji Kamiyama, and Hiroaki Kariyazaki

Dept. of Communication Eng., Okayama Prefectural University, Japan1

Introduction

In 1982, Voronkov presented a model describing point
defect behavior during the growth of single crystal Si from amelt and derived an expression to predict if the crystal was
vacancy- or self-interstitial-rich [1]. Recently, Vanhellemontclaimed that one should take into account the impact ofcompressive stress introduced by the thermal gradient at themelt/solid interface by considering the hydrostatic pressure
dependence of the formation enthalpy of the intrinsic point
defects [2]. To evaluate the impact of thermal stress morecorrectly, the pressure dependence of both the formation
enthalpy (Hf) and the activation enthalpy (Hm) of the intrinsic
point defects should be taken into account. Furthermore,
growing single crystal Si is not under hydrostatic pressure but
almost free of external pressure (generally in Ar gas under
reduced pressure).
In the present paper, the dependence of Hf and Hm on thepressure P, or in other words, the pressure dependence of theformation energy (Ef) and the relaxation volume (vf), isquantified by density functional theory calculations. Although alarge number of ab initio calculations of the properties ofintrinsic point defects have been published during the last years,calculations for Si crystals under pressure are rather scarce [3].
For vacancies V, the reported pressure dependences of HfV areinconsistent [4,5].
The thermal stress in growing single crystal Si iscompressive, and is not external but internal stress. Therefore,
the point defects are under internal pressure. To estimate theimpact of thermal stress on intrinsic point defects in growing Si,
the differences in the formation volumes of point defects underhydrostatic pressure and under internal pressure should beconsidered [6]. Taking this into account, we estimated theimpact of thermal stress on Γ0crit by using DFT results.
2

Calculation Details

In the present work, DFT calculations were performed
within the generalized gradient approximation (GGA) forelectron exchange and correlation, using the CASTEP code [7].
The wave functions were expanded with the plane waves, andthe ultra-soft pseudo-potential method was used to reduce thenumber of plane waves. The cutoff energy was 340 eV. The
expression proposed by Perdew et al. [8] was used for theexchange-correlation energy in the GGA. Periodic boundary
conditions were used with cubic supercells of 216 atoms forcalculations of perfect and defect-containing Si crystals. k-point
sampling was performed at 2×2×2 special points in a
Monkhorst-Pack grid [9].
The reference point in this study was the perfect Si crystal.
The pressure P in cubic cells with different volumes wasdetermined by fitting the total energy for different volumes to a
Birch-Murnaghan equation of state [10]. The pressures P in
cubic cells with different volumes were also obtained
analytically with the method by Nielsen and Martin [11].
Similar calculations were performed with cubic supercells
containing point defects. In single crystal Si, the thermal

equilibrium concentrations of self-interstitials I and vacancies V
even near melt temperature, are well below 5×1015 cm-3. Forsuch low concentration, the calculation cells should besurrounded by perfect cells. These perfect Si crystals should
deform isotropically under the hydrostatic pressure. Therefore,
we imposed the cubic shape for the defect-involving calculation
cell. In each case, the supercells were set at a particular cubic
volume and the ionic coordinates were fully relaxed to build up
a list of energy-pressure-volume data points by using theanalytically obtained P. In the present study only neutral point
defects were considered.
For the self-interstitial, the two lowest energy
configurations, i.e. the [110] dumbbell (D) and the tetrahedral(T) sites were calculated. It is well known that, for neutral I, the[110] D-site is the lowest-energy configuration while the T-siteis the transition state. A vacancy was introduced by eliminating
one Si atom located around the center of each supercell. Further
details of the point defect configurations considered in this study
will be commented on later. The energy-pressure-volume dataof perfect and defect-involving cubic cells were used to find therelationship between pressure P and (1) formation energy Ef, (2)relaxation volume vf, and (3) formation enthalpy Hf.
3Results and discussion

In case of the hydrostatic pressure Ph, Fig. 1(above) shows
the calculated formation enthalpy HfI at the [110] dumbbell site
(most stable). For self-interstitials, HfI and HmI are given by HfI =3.425 – 0.057×Ph (eV) and HmI = 0.981 – 0.039×Ph (eV),respectively, with Ph given in GPa. Fig. 1(below) shows thecalculated formation enthalpy HfV for Jahn-Teller distortion (hJT, most stable) of vacancies V. The calculated HfV and HmV
dependencies on Ph given by HfV =3.543 – 0.021×Ph2 – 0.019×Ph(eV) and HmV = 0.249 + 0.018×Ph2 – 0.037×Ph (eV),respectively. These results indicate that, when assuming that thepre-factors in the Arrhenius equation are not influenced,hydrostatic pressure up to 1 GPa leads to a slight increase of thethermal equilibrium concentration and diffusion of vacancies
but this increase is much smaller than that of self-interstitials.
In case of the internal pressure Pin, Fig. 2 shows thechanges in formation enthalpies ∆HfI for D-site (most stable) andT-site (transition state) of self-interstitials I, and ∆HfV for JahnTeller distortion (h-JT, most stable) and split-V (transition state)of vacancies V due to internal pressure Pin. We found that the HfIof I increases while the HfV of V decreases with the increase inPin. The calculations lead to the dependencies of HfI, HmI and HfV,
HmV on Pin given by HfI = 3.425 + 0.070×Pin (eV) and HmI =0.981 – 0.038×Pin (eV), respectively, with Pin given in GPa. The
calculated HfV and HmV dependencies on Pin given by HfV =3.543– 0.160×Pin (eV) and HmV = 0.249 – 0.026×Pin (eV),respectively. These results indicate that compressive thermalstress leads to increases in the equilibrium concentration anddiffusion of V and to a decrease in the equilibrium concentrationof I. By the recombination between V and I, compressive
thermal stress makes Si crystals more V-rich.

Fig. 3 Variations in critical v/G (v: pulling speed, G: temperaturegradient) ratio Γ0crit with thermal stress σ0 = Pin. Results
obtained by using values for Hf and Hm under hydrostaticpressure Ph are also given.
4Fig.1 Dependence of formation enthalpy HfI for D-site (above)and HfV for h-JT (below) on the hydrostatic pressure Ph.

Conclusion

The dependences of the formation enthalpy (Hf) and theactivation enthalpy (Hm) of the self-interstitial I and the vacancy
V on the hydrostatic pressure Ph and on the internal pressure Pin,
were calculated by calculating the formation energy (Ef) andrelaxation volume (vf).
The neutral I and V are found to have quasi constantformation energies EfI and EfV for pressures up to 1 GPa. For therelaxation volume, vfI is almost constant while vfV decreases
linearly with increasing pressure P. The dependencies offormation enthalpies HfI for D- and T-sites, HfV for h-JT (moststable) and split-V (transition state) on internal pressure Pin wereobtained. We found that the HfI of I increases while the HfV of V
decreases with internal pressure Pin.
The results we obtained were used to more accurately
describe the impact of thermal stress on Γ0crit than the estimates
under hydrostatic pressure. A nearly linear relation between
Γ0crit (in 10-3 cm2min–1K–1) and σ0 (in MPa) was obtained,described by Γ0crit ≈ 1.509–0.023σ0. The impact of thermal
stress on Γ0crit makes the growing Si crystal more vacancy-rich.
The results illustrate that it is important to take into account theimpact of stress on the generation of intrinsic point defects indeveloping future large diameter defect-free crystals.
Acknowledgements
Jan Vanhellemont (Ghent University) is acknowledged forvaluable discussion through this work. Kozo Nakamura(SUMCO Corporation) is acknowledged for the discussion on
the formation volumes of point defects in prefect crystal Si.
ReferencesFig. 2 Changes in formation enthalpies ∆HfI for D-site (moststable) and T-site (transition state), ∆HfV for Jahn-Teller
distortion (h-JT, most stable) and split-V (transition state) due tointernal pressure Pin.
The obtained result for internal pressure was used to moreaccurately describe the impact of thermal stress on Voronkov
criterion [1] Γ0crit than the estimates under hydrostatic pressure
as shown in Fig. 3. A nearly linear relation between Γ0crit (in 103
cm2min–1K–1) and the thermal stress at the melt/solid interface
σ0 (in MPa) was obtained, described by Γ0crit ≈ 1.509–0.023σ0.
The impact of thermal stress on Γ0crit makes the growing Si
crystal more V-rich. The results illustrate that it is important totake into account the impact of stress on the generation of
intrinsic point defects in developing future large diameter
defect-free crystals.

[1] V. Voronkov, J. Cryst. Growth 59 (1982) 625.
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110 (2011) 129903.
[3] B. Puchala, PhD Thesis, Table 2.1 and 2.2, The Universityof Michigan (2009) and references therein.
[4] S. Centoni et al., Phys. Rev. B72 (2005) 195206.
[5] M. Ganchenkova et al., Nucl. Inst. Meth. Phys. Rev. B 202(2003) 107.
[6] M. Aziz, Materials Science in Semiconductor Processing 4(2001) 397, and references therein.
[7] The CASTEP code is available from Accelrys Software Inc.
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[9] H. Monkhorst and J. Pack, Phys. Rev. B13 (1976) 5188.
[10] J. Poirier, Introduction to the Physics of the Earth’sInterior, 2nd ed., Cambridge University Press, New York,(2000).
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