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Three-dimensional analysis of Marangoni flow and radial segregation in Ge x Si1-x melt with Czochralski configuration

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Abstract

In order to examine the flow field and the radial segregation of silicon (Si) in a Ge x Si1-x melt with an idealized Czochralski (Cz) configuration, we conducted a series of unsteady three-dimensional (3-D) numerical simulations under zero-gravity conditions. The effect of convection driven by surface tension on the free surface of the melt was included in the model, by considering thermal, as well as solutal Marangoni convection. The concentration and flow fields at several stages during crystal growth are presented for several temperature differences, driving the Marangoni convection. The simulation results indicate that the flow and concentration fields are axisymmetric for Ma T <  625 and become oscillatory and 3-D for higher values. It was found that the maximum Si concentration difference at the growth interface decreases as thermal Marangoni number increases due to higher flow velocities in the vicinity of the interface. However, temporal fluctuations of Si concentration at the interface increase at higher thermal Marangoni numbers. The effects of aspect ratio (A r) were also considered in the model. It was found that the aspect ratio of the melt in the crucible has a prominent influence on the flow pattern in the melt which, in turn, effects the Si concentration at the growth interface.

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Abbreviations

A r :

aspect ratio, H/r cru

Bi rad :

Biot number for radiation heat transfer, \({\frac{{r_{{\rm cru}}}}{k}\varepsilon \sigma {\left({T + T_{a}} \right)}{\left({T^{2} + T^{2}_{a}} \right)}}\)

C :

dimensionless molar fraction of Si in the melt, C */C * o

C * o :

initial molar fraction of Si in the melt

C * :

molar fraction of Si in the melt

C P :

specific heat capacity, J kg−1 K−1

D :

diffusion coefficient, m2 s−1

ΔT :

reference temperature difference, T h − T m

H :

crucible height, m

k :

thermal conductivity, W m−1 K−1

k s :

segregation coefficient

Ma T :

thermal Marangoni number, − ∂γ/∂T(ΔTr cru/μα)

Ma C :

solutal Marangoni number, − ∂γ/∂C *(C * o r cru/μα)

P :

dimensionless pressure

Pr :

Prandtl number, ν/α

R :

dimensionless radial distance, r/r cru

R cry :

dimensionless crystal radius, r cry/r cru

r :

radial distance, m

Sc :

Schmidt number, ν/D

U :

dimensionless radial velocity component.

V :

dimensionless circumferential velocity component

W :

dimensionless vertical velocity component

V L :

crystal growth rate, m s−1

r cru :

crucible radius, m

T :

temperature K

t :

dimensionless time

∂γ/∂T :

temperature coefficient of surface tension, N m− 1 K−1

∂γ/∂C * :

concentration coefficient of surface tension, N m−1

z :

vertical distance, m

Z :

dimensionless vertical distance, z/r cru

α:

thermal diffusivity, m2 s−1

σ:

Stefan-Boltzman constant, = 5.67040 × 10− 8 W m−2 K−4

Θ:

dimensionless temperature, (T − T m)/ΔT

Θ a :

dimensionless ambient temperature, =  (T a − T m)/ΔT

ε:

emmisivity

μ:

dynamic viscosity, kg m− 1 s−1

ν:

kinematic viscosity, m2 s−1

θ:

circumferential direction, rad

ρ:

density, kg m−3

a:

ambient

C :

solutal

cru:

crucible

cry:

crystal

h:

heated wall

m:

melting

T :

thermal

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Authors and Affiliations

  1. Mechanical Engineering Department, Eastern Mediterranean University, Magosa, Mersin 10, Turkey

    S. Abbasoglu & I. Sezai

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  1. S. Abbasoglu
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Abbasoglu, S., Sezai, I. Three-dimensional analysis of Marangoni flow and radial segregation in Ge x Si1-x melt with Czochralski configuration. Engineering with Computers 23, 123–135 (2007). https://doi.org/10.1007/s00366-006-0052-8

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  • DOI: https://doi.org/10.1007/s00366-006-0052-8

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