Author's Accepted Manuscript

Efficient computation of population distribu-tion of microdefects at any location in grow-ing Czochralski silicon single crystals

Gaurab Samanta, Milind Kulkarni

PII: S0022-0248(13)00680-5DOI: http://dx.doi.org/10.1016/j.jcrysgro.2013.10.018Reference: CRYS21800

To appear in: Journal of Crystal Growth

Received date: 8 August 2013Revised date: 8 October 2013Accepted date: 9 October 2013

Cite this article as: Gaurab Samanta, Milind Kulkarni, Efficient computation ofpopulation distribution of microdefects at any location in growing Czochralskisilicon single crystals, Journal of Crystal Growth, http://dx.doi.org/10.1016/j.jcrysgro.2013.10.018

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Efficient computation of population distribution of microdefects at any location in growing Czochralski silicon single crystals

Gaurab Samanta∗, and Milind Kulkarni Quantitative Silicon Research, SunEdison Inc., St. Peters, MO 63376, USA

Date: 10/07/2013 ∗Corresponding author. Tel.: +1 30 2561 0832. gaurab.samanta@gmail.com Abstract A computationally efficient model to quantify the distribution of microdefect population at any given location in Czochralski grown silicon crystals is proposed and numerically solved. For this purpose, all microdefects are approximated as spherical clusters and the concentration fields of intrinsic point defects and other intermediate species are evolved using a lumped model which eliminates the requirement to store formation and path histories of the clusters. The formation of all clusters is modeled by the classical nucleation theory, while they grow by diffusion-limited kinetics. The model is validated by comparing its predictions against that of a rigorous model, where actual cluster population distribution is captured on the basis of formation and path histories of the clusters. PACS: 81.10.Fq; 81.10.Aj; 81.05.Hd; 61.72.Yx Keywords: A1. Defects; A1. Microdefects; A1. Point defects; A1. Nucleation; A1. Size distribution; A2. Czochralski method; B2. Semiconducting silicon; 1. Introduction Silicon crystals grown by Czochralski (CZ) process contain microdefects, resulting commonly from aggregations of intrinsic point defects and oxygen. Good quantitative predictions of microdefect distribution significantly enhance the efficacy with which the crystal growth process can be tailored to obtain desired defect characteristics. Using a continuum approach based on classical nucleation theory, a rigorous quantitative model was proposed [1]. In the continuum description of defects physics, population of aggregates of varying sizes exists at all locations within the crystal. Therefore, the rigorous model capturing population of microdefects at every crystal location becomes computationally very expensive, especially when it is extended to two dimensions. In [2], a lumped representation of the microdefect population is used to devise an algorithm which is computationally efficient yet physically accurately captures the radial as well as axial distributions of average sizes of microdefects in a CZ crystal. However, the computational advantage came at the expense of the information of the distribution of the clusters within a population at any given location.

In a segment of a growing CZ crystal, defect clusters nucleate at different times and temperatures which grow to different sizes. Hence, most of the computational expenditure of the rigorous model goes into keeping track of formation and path histories of various aggregates or clusters. In this paper, based on the model described in [2], a method is presented which allows for calculation of size distribution of microdefect population at any location in the crystal

without expending on formation and path histories of the clusters. Knowledge of size distribution of microdefects population is necessary to estimate density of bulk microdefects (BMD) in an as-grown crystal. Presence of BMD in sufficient quantity in the bulk of the CZ wafer ensures good removal of device-degrading impurities i.e effective gettering. Therefore, size distributions of microdefects provide a way to quantify gettering capacity of an as-grown wafer. 2. Model Development 2.1 Governing equations Primarily, CZ defect dynamics is influenced by the Frenkel reaction, which involves the mutual annihilation of a vacancy and a self-interstitial to produce a silicon lattice atom, and the generation of a pair of a vacancy and a self-interstitial from a lattice silicon atom. The net rate of the recombination or annihilation of the point defects by the Frenkel reaction per unit volume, r , is given as

( ), ,iv i v i e v er k C C C C= − , (1)

where ivk is the recombination rate constant, and C is the point defect concentration. While the subscripts i and v denote interstitials and vacancies, respectively, e refer to equilibrium conditions. Since the Frenkel reaction can be assumed to be much faster than the diffusive or convective modes of transport of point defects, the following relationship remains valid,

, ,i v i e v eC C C C= . (2) The concentrations of point defect species are assumed to exist at their equilibrium values at the melt-crystal interface. Beyond a region close to the interface, known as the recombination length [3], a series of bimolecular reactions can lead to nucleation of point defect clusters which can be treated as spherical bodies. The total free energy change associated with the formation of a cluster at a given temperature and point defect concentration containing m point defects,

( )x xF m , is given by [4-6]

{ }2/3

,

( ) ln ,xx x x b x x

x e

CF m m k T m x i vC

λ⎛ ⎞

= − + =⎜ ⎟⎜ ⎟⎝ ⎠

, (3)

where bk is the Boltzmann constant, T is the temperature, λ is the surface energy coefficient for the new cluster. The rate of formation of stable nuclei or clusters, J , per unit volume as given by classical nucleation theory is [5-7]

( ) { }*)(

1/2* *,

,

4 ( ) 12 ( ) ln ,x x

b

F mk Tx

x x x x x x x b b site xx e

CJ R m D C F m k T k T e x i vC

π π ρ−

−⎡ ⎤⎛ ⎞⎛ ⎞⎡ ⎤ ⎢ ⎥= =⎜ ⎟⎜ ⎟⎣ ⎦ ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

, (4)

where R is the cluster radius, D is the diffusivity of the point defects, siteρ is the site density for nucleation. The superscript ‘*’ denotes critical clusters. The subscript x refers to point defects as well as clusters of point defects depending on the variable. The stable clusters need to

overcome a high energy barrier to form. Therefore, once formed, the stable clusters can be assumed to grow by diffusion-limited kinetics.

As described previously in [2, 7, 8], the density and a representative size of stable clusters can be solved explicitly as

{ },x xx

n nV J x i vt z

∂ ∂+ = =

∂ ∂, (5)

( ) { },2 ,x x x x

x x exx

U U D nV C C x i vt z ψ

∂ ∂+ = − =

∂ ∂, (6)

1

1 22 2 x

x xx

URn

⎛ ⎞ℜ = = ⎜ ⎟

⎝ ⎠, (7)

where xn is the number of clusters per unit volume, V is the crystal pull rate, t is time and z is the direction in which the crystal is pulled. U is related to the representative radius xℜ of the cluster population which is square root of the average of the squares of the radii of all clusters in the population as shown in equation (7). In other words, the lumped size parameter represents the surface area of the cluster population. x

xψ is the density of monomer x in a cluster of point defects of the same type, which is assumed to be equal to the lattice site density.

The equation describing the balance of the excess vacancy concentration, defined as the

difference between the concentrations of vacancies and self-interstitials, involving their transport and consumption by Frenkel reaction and the existing clusters is ( ) ( ) ( ) ( )v i v i v i

v v i i v i

C C C CV D C D C q q

t z∂ − ∂ −

+ = ∇ ⋅ ∇ −∇ ⋅ ∇ − +∂ ∂

. (8)

where the consumption of the point defects in the formation of stable nuclei is neglected. v

vq and i

iq represent the volumetric consumption rates of vacancies by vacancy clusters and self-interstitials by clusters of self-interstitials, respectively. These volumetric consumption rates can be assumed to be diffusion-limited as described in [2, 7, 8],

( ) { },4 ,xx x x x e x xq D C C n x i vπ= − ℜ = . (9)

2.2 Size distribution calculation In order to obtain the distribution of the cluster population at any location in the crystal, the knowledge of number density of clusters of all sizes in the population is required. For this purpose, it is proposed that the increment in number density or the newly formed stable supercritical clusters at a particular location at different times be tracked and stored separately in a variable denoting a fraction of the population

( ( ))t t

p xx p

t

nn R t dtt

+Δ ∂⎛ ⎞Δ = ⎜ ⎟∂⎝ ⎠∫ , (10)

whose size ( )pR t is then evolved in time using the diffusion-limited growth equation

( ) { }2 2

,2 ,p p x

x x exx

R R DV C C x i vt z ψ

∂ ∂+ = − =

∂ ∂. (11)

Since the initial size of a stable cluster is negligible compared to its final size, it can be assumed to be zero. A schematic representation of the procedure of evaluating population distribution in lumped model is given in Fig. 1. The development of cluster population at one chosen location, denoted by the horizontal line in the crystal, is done by building a table of population fractions and their corresponding radii. Thus, a histogram of the underlying size distribution of various clusters at any location can be obtained by plotting p

xnΔ versus pR . Utilizing this information to calculate the derivative of the number density with respect to radius of cluster and normalizing it by the total number density of all clusters, the size distribution of clusters at any location can be obtained. When the formation and growth of clusters seize this procedure gives the mature size distribution of the clusters.

In the aforementioned modeling framework, there is still no requirement to record the histories of nucleation rates at all elapsed times. Also, the size distribution calculation can be restricted to only some pre-selected locations in the crystal. Therefore, the several factors of anticipated gain in computational time of the lumped model over the rigorous model [2] should still remain valid. An estimate of the time savings of the lumped algorithm is 2( )tO N where tN is the number of points in the time domain. The number tN depends on crystal growth conditions such as pull rate and length of the crystal. For example, to grow a crystal of length 1 m at a pull rate of 0.5 mm/min, typically tN would be about 310 .

Although the treatment here has been solely concerned with the cluster size distribution of vacancy and interstitial clusters, it can be easily applied to oxygen precipitates. Moreover, the modeling approach will also not change when nitrogen is present. The governing equations of the lumped model for defect dynamics in the presence of oxygen and nitrogen are already published elsewhere [9, 10]. The computational domain is a function of time depicting the evolution of the crystal geometry. On all crystal surfaces, the point defect concentrations are assumed to be at equilibrium at the local temperature. The time scale of heat transport in the crystal is much smaller than that of defect dynamics. Hence, the model can be solved for any given steady state temperature field. Except for vacancies and self-interstitials all species are assumed to be immobile.

Earlier, the representative radius of the cluster population obtained from lumped model

was shown to agree well with the average radius of the clusters calculated from rigorous model [2]. Two-dimensional calculation of the representative sizes of defect clusters in a CZ crystal using the lumped model were also found to be in close agreement with experimental observations [2]. In the remainder of this article, comparisons of cluster size distributions in a growing CZ crystal obtained using the lumped model are made with those obtained using the rigorous model [1]. For this purpose, the results reported in [1] using the rigorous model have been directly used while making sure that the lumped model is being run for the same conditions as in [1]. 3. Comparison between the rigorous and lumped models 3.1 Steady state simulations Since temperature fields are assumed to remain fixed, steady state refers to crystal pulling at fixed rate so that crystal growth proceeds steadily in time. For the axial temperature profile and the point defects property values provided in [1], microdefect size distributions in CZ crystals were computed for two fixed pull rates using the lumped model. The size distributions corresponding to rigorous model were obtained from [1]. Corresponding to the pull rate values reported in [1], a pull rate of 0.225 mm/min was chosen to study an interstitial-rich crystal while a pull rate of 0.6 mm/min was chosen to investigate a vacancy-rich crystal. In Figs. 2 and 3, the size distributions of mature clusters (i.e. after their complete growth) are shown. The agreement between the results from rigorous and lumped model is reasonably good. The n th moment of the size distribution function gives the expected value of n

xR . The first moment which gives the mean of the cluster sizes xR features in the rigorous expression of the total consumption rate of the point defects by the clusters per unit volume [1]. The second

moment which is the area-averaged radius 1/22

xR of the cluster distribution and the third

moment with the meaning of volume-averaged radius 1/33

xR , both have been used previously as representatives of the actual cluster sizes [1, 2, 10, 11]. In fact, the cluster sizes are lumped into the second moment when developing the lumped model to describe the formation and growth of microdefects. All three moments of the size distributions shown in Figs. 2 and 3 are reported in table 1. The v -cluster size averages are slightly overestimated, and exhibit larger relative error compared to the size averages of i -clusters.

Some differences in the values are expected because the volumetric consumption rates calculated in the lumped model are based on root mean square rather than the mean size of the total cluster population. Vacancy clusters which are formed larger in number and smaller in size have larger surface area to volume ratio compared to the clusters of self-interstitials. Therefore, in that respect, a root mean square representation of the size distribution is more appropriate for the v -clusters, albeit shifting the distribution towards bigger radius because of overestimation of volumetric consumption rate. On the other hand, for i -clusters, the overestimation of volumetric consumption rates in the lumped model due to the use of root mean square value of size rather than the mean size is not as pronounced because of their higher diffusivity value at higher temperatures. As a result, the relative error in size averages calculated based on the size

distribution achieved by the lumped model is significantly lower. The very small underestimation seen in the values for i -clusters could be numerical error.

3.2 Unsteady state simulations The same temperature profile and the property values discussed in the previous section are used for an unsteady state crystal growth simulation of defect dynamics where only axial variation is considered. The unsteady pull rate profile shown in Fig. 16 of [4] is used. For brevity, this figure is not shown here.

The comparison between the rigorous and lumped model based calculations of the size distributions of defect cluster population at different axial locations along the crystal is shown in Fig. 4. The spatial variation in the microdefect size distribution because of the effect of unsteady state conditions is well reproduced by the lumped model. An explanation of the unsteady state effects can be found in [1]. Similar to the steady state cases, the size distributions of vacancy clusters as obtained from lumped model are found to be shifted towards higher values of cluster radius compared to the rigorous model. 4. Discussion The parameter set used for the calculations here is the same as provided in [1]. The dependences of thermal stress and dopant concentration are not taken into account. The dopant-induced strain effect on intrinsic point defect equilibrium concentrations is negligibly small for conventional doping levels [11, 12]. The change in equilibrium values of point defect concentrations due to dopant-induced shift in the Fermi level is also significant only at very high doping levels (of the order of 1810 cm-3 or more) [11, 13]. The impact of thermal stress on equilibrium concentration values is also found to be small [12]. The formation and migration energies of vacancies and self-interstitials are dependent on the thermal stress [12]. The implication of this effect on critical (V/G) at the interface is also discussed in [12]. However, our aim in the present study is simply to delineate and validate a method to obtain size distribution of defect clusters in the lumped model approach. Therefore, accounting for the dependency of formation and migration energies of point defects on thermal stress is not required in the scope of this work. The lumped model uses the scaling dictated by the diffusion-limited growth equation to choose root mean square size as the representative of defect cluster population. While using volumetric mean size to represent the population will be more accurate way of lumping, it will make the governing equation for growth more complicated. It is deemed unnecessary at this point to delve in such complication given the good agreement of size distributions and statistics thereof between lumped and rigorous models as shown in Figs. 2-4 and table 1.

Experimental validation of number densities of point defect clusters obtained from rigorous model has been done before [1]. The as-grown size distribution data of voids or self-interstitial clusters is difficult to obtain experimentally. In lieu of experimentally measured size distributions, comparisons are made with rigorous model here. However, it is noted that Akatsuka et. al. [14] reported as-grown size distribution of voids determined experimentally

using an optical precipitate profiler. Hence, in future, it is of interest to pursue a direct comparison against experimental measurements.

5. Conclusions The computational advantage of the lumped model is not compromised when the point defect concentration field evolved through it is also used to solve for the size distribution of microdefect populations at chosen locations in the crystal. The capability to quantify size distribution of microdefect population in a time-efficient manner is a significant accomplishment as it enables predictions on BMD densities from crystal growth conditions. The size distributions predicted from lumped model agree reasonably well with those predicted from rigorous model under steady as well as unsteady conditions. 6. References

1. M. S. Kulkarni, V. V. Voronkov, and R. Falster, “Quantification of defect dynamics in unsteady-state and steady-state Czochralski growth of monocrystalline silicon”, J. Electrochem. Soc., 151, G663-G678 (2004).

2. M. S. Kulkarni, and V. V. Voronkov, “Simplified two-dimensional quantification of the grown-in microdefect distributions in Czochralski grown silicon crystals”, J. Electrochem. Soc., 152, G781-G786 (2005).

3. V. V. Voronkov, “The mechanism of swirl defects formation in silicon”, J. Crystal Growth, 59, 625-643 (1982).

4. D. Turnbull, and J. C. Fisher, “Rate of nucleation in condensed systems”, J. Chem. Phys., 17, 71 (1949).

5. A. S. Michaels, “Nucleation Phenomena”, American Chemical Society, Washington DC (1966).

6. D. Kashchiev, “Nucleation Basic Theory with Applications”, Butterworth-Heinemann, Woburn, MA (2000).

7. M. S. Kulkarni, “Continuum-scale quantitative defect dynamics in growing Czochralski silicon crystals”, in: Springer handbook of crystal growth, part F, 1281-1334, Springer Berlin Heidelberg (2010).

8. V. V. Voronkov, B. Dai, and M. S. Kulkarni, “Fundamentals and engineering of the Czochralski growth of semiconductor silicon crystals” in: P. Bhattacharya, R. Fornari, and H. Kamimura (eds.), Comprehensive semiconductor science and technology, 3, 81-169, Amsterdam: Elsevier (2011).

9. M. S. Kulkarni, “Defect dynamics in the presence of oxygen in growing Czochralski silicon crystals”, J. Crystal Growth, 303, 438-448 (2007).

10. M. S. Kulkarni, “Defect dynamics in the presence of nitrogen in growing Czochralski silicon crystals”, J. Crystal Growth, 310, 324-335 (2008).

11. V. V. Voronkov, and R. Falster, “Effect of doping on point defect incorporation during silicon growth”, Microelectron. Eng., 56, 165-168 (2001).

12. K. Sueoka, E. Kamiyama, and J. Vanhellemont, “Theoretical study of the impact of stress on the behavior of intrinsic point defects in large-diameter defect-free Si crystals”, J. Crystal Growth, 363, 97-104 (2013).

13. V. V. Voronkov, and R. Falster, “Dopant effect on point defect incorporation into growing silicon crystal”, J. Appl. Phys., 87, 4126-4129 (2000).

14. M. Akatsuka, M. Okui, S. Umeno, and K. Sueoka, “Calculation of size distribution of void defects in CZ silicon”, J. Electrochem. Soc., 150, G587-G590 (2003).

Table 1: Comparison of different means of radii of i − clusters and v − clusters whose size distributions are shown in Figs. 2 and 3, respectively Model

vR (nm) 1/22vR

(nm)

1/33vR

(nm)

iR (nm) 1/22iR

(nm)

1/33iR

(nm) Rigorous 62.91 63.12 63.32 225.48 227.15 228.80 Lumped 67.0 67.45 67.72 221.95 223.42 225.05

Fig. 1: Schematic description of the algorithm followed to calculate cluster population distribution from lumped model.

 Fig. 2: Comparison between rigorous and lumped models, of mature i-cluster size distribution in a simulated interstitial rich crystal grown under steady state at pull rate of 0.225 mm/min.

Fig. 3: Comparison between rigorous and lumped models, of mature v-cluster size distribution in a simulated vacancy rich crystal grown under steady state at pull rate of 0.6 mm/min.

Fig. 4: Comparison between rigorous and lumped models, of v-cluster size distribution in a simulated vacancy rich crystal grown under unsteady state. The pull rate profile used here is shown in Fig. 16 of [1].