Dopant transport during semiconductor crystal growth

Hirtz, Julie M.; Ma, Nancy

doi:10.1016/s0022-0248(99)00815-5

Cited by 18 publications

(4 citation statements)

References 10 publications

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“…Substituting the above expansion back into Eq. (14) and comparing the terms with equal powers of p we obtain f j ðaÞ ¼ P j l¼0 f l ðaÞg l;jÀl ðaÞ; that due to g l;0 ¼ 1 simplifies to P jÀ1 l¼0 f l ðaÞg l;jÀl ðaÞ ¼ 0: Upon replacing j by j þ 1 and taking into account g j;1 ðaÞ ¼ Àj, the latter relation results in…”

Section: Solution By Laplace Transformmentioning

confidence: 99%

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Breakdown of Burton–Prim–Slichter approach and lateral solute segregation in radially converging flows

Priede

Gerbeth

2005

Journal of Crystal Growth

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A theoretical study is presented of the effect of a radially converging melt flow, which is directed away from the solidification front, on the radial solute segregation in simple solidification models. We show that the classical Burton-Prim-Slichter (BPS) solution describing the effect of a diverging flow on the solute incorporation into the solidifying material breaks down for the flows converging along the solidification front. The breakdown is caused by a divergence of the integral defining the effective boundary layer thickness which is the basic concept of the BPS theory. Although such a divergence can formally be avoided by restricting the axial extension of the melt to a layer of finite height, radially uniform solute distributions are possible only for weak melt flows with an axial velocity away from the solidification front comparable to the growth rate. There is a critical melt velocity for each growth rate at which the solution passes through a singularity and becomes physically inconsistent for stronger melt flows. To resolve these inconsistencies we consider a solidification front presented by a disk of finite radius R 0 subject to a strong converging melt flow and obtain an analytic solution showing that the radial solute concentration depends on the radius r as $ln 1=3 ðR 0 =rÞ and $ lnðR 0 =rÞ close to the rim and at large distances from it. The logarithmic increase of concentration is limited in the vicinity of the symmetry axis by the diffusion becoming effective at a distance comparable to the characteristic thickness of the solute boundary layer. The converging flow causes a solute pile-up forming a logarithmic concentration peak at the symmetry axis which might be an undesirable feature for crystal growth processes. r

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Section: Solution By Laplace Transformmentioning

confidence: 99%

“…The simulation of dopant transport is an important aspect of crystal growth modelling [14,15], and various numerical approaches are used for it. However, a numerical approach is always limited in the sense that it provides only particular solutions while the basic relations may remain hidden.…”

Section: Introductionmentioning

confidence: 99%

Breakdown of Burton–Prim–Slichter approach and lateral solute segregation in radially converging flows

Priede

Gerbeth

2005

Journal of Crystal Growth

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“…[20][21][22][23][24][25]. These authors use a mathematical approach similar to matched asymptotic expansions, in which a numerical solution to an inviscid flow problem in a core region is matched to asymptotic solutions within the thin viscous Hartmann layers at the boundaries.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of three-dimensional convection in microgravity crystal growth: g-jitter with steady magnetic fields

Yeckel

Derby

2004

Journal of Crystal Growth

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“…For example, all physical properties were treated as constants instead of as functions of temperature. Transport equation for the passive scalar (concentration of the microfibers) was not included, but could be added relatively easily [16]. Its addition would enable us to predict and possibly control the distribution of the microfibers along the magnetic field lines of force.…”

Section: Discussionmentioning

confidence: 99%

An Inverse Method Allowing User-Specified Layouts of Magnetized Microfibers in Solidifying Composites

Dulikravich

Colaço

Martin

et al. 2003

Journal of Composite Materials

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A new concept for controlled manufacturing of specialty composite materials has been developed. It is based on the concept of aligning magnetizable microfibers tangent to magnetic field lines of force. This paper offers a method that allows users to specify desired patterns of the magnetic lines of force that can be achieved by an appropriate positioning of magnets of various strengths along the boundaries of the composites' curing container. An improved analytical model and a numerical algorithm have been developed for the prediction of the magnetic force lines inside a flowing and solidifying melt. This analysis code was combined with a hybrid constrained optimization algorithm that minimizes a normalized sum of least square differences between the user-specified and the predicted patterns of the magnetic lines of force. This software package was used to inversely evaluate the strengths, locations, and orientations of magnets needed to generate the user-specified patterns of the magnetic lines of force in the solidifying composite thus verifying the conceptual feasibility of this novel manufacturing process.

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Dopant transport during semiconductor crystal growth

Cited by 18 publications

References 10 publications

Breakdown of Burton–Prim–Slichter approach and lateral solute segregation in radially converging flows

Breakdown of Burton–Prim–Slichter approach and lateral solute segregation in radially converging flows

Dynamics of three-dimensional convection in microgravity crystal growth: g-jitter with steady magnetic fields

An Inverse Method Allowing User-Specified Layouts of Magnetized Microfibers in Solidifying Composites

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