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1、全球半导体晶体生长建模著名商业软件FEMAG Global Simulation of Czochralski Silicon Growth under the Effect of a Transverse Magnetic Field,FEMAGSoft 2013,Cz Si growth under a TMF,MHD boundary layers: Hartmann layers,FEMAGSoft 2013,The Hartmann layers develop along the surfaces where the normal component of the magnetic
2、 field is non-negligible.,An order of magnitude of the thickness of a Hartmann layer is dH = L Ha-1 (L = Rs or Rc). Typically dH = 0.05 - 0.08 mm in industrial furnaces.,Cz Si growth under a TMF (contd),Transverse magnetic fields: FLET method,Method: Fourier decomposition of all fields (velocity, pr
3、essure, temperature),Hypothesis:,Principal issue: which and how many modes ?,Objective: global, quasi-steady or time-dependent calculations at a reasonable cost,FEMAGSoft 2013,Cz Si growth under a TMF (contd),Fourier Limited Expansion Technique (FLET),FEMAGSoft 2013,The field variables are expanded
4、as Fourier series in the azimuthal (q) direction:,Cz Si growth under a TMF (contd),Fourier Limited Expansion Technique (FLET),FEMAGSoft 2013,and while the number of modes M is as small as possible without loss of accuracy (spectral convergence).,while the different mode coefficients:,are 2D Finite E
5、lement functions of the meridional coordinates (r, z),This results in a system whose size is that of the 2D system multiplied by the number of Fourier modes considered and hence in a dramatic system size reduction.,Cz Si growth under a TMF (contd),FEMAGSoft 2013,Radiation transfer,Coupling between 3
6、D and 2D axisymmetric heat transfer across a radiative enclosure,the viewed and hidden parts are calculated as axisymmetric or, equivalently, each surface of the enclosure is viewed as axisymmetric from the other surfaces,Main modeling hypothesis:,generally 3D components are rotating with respect to
7、 the 2D environment 3D components mostly view 2D components because of the presence of heat shields,This hypothesis is satisfactory because:,Cz Si growth under a TMF (contd),FEMAGSoft 2013,Flow and global heat transfer in a silicon Cz puller under the effect of a TMF (quasi-steady simulation),Cz Si
8、growth under a TMF (contd),FEMAGSoft 2013,Top: top view of the velocity magnitude and streamlines. Bottom: velocity field magnitude and cross-section showing a sharp Hartmann layer along the melt-crucible interface.,Growth of a 300 mm diameter Si crystal under the effect of a 0.5 T TMF.,Cz Si growth
9、 under a TMF (contd),FEMAGSoft 2013,In a typical TMF configuration extremely thin Hartmann layers of 50-80 mm develop,Growth of a 300 mm diameter Si crystal under the effect of a 0.5 T TMF.,Cz Si growth under a TMF (contd),FEMAGSoft 2013,Detail of the deforming Boundary Layer Mesh (BLM) used,Flow an
10、d global heat transfer in a silicon Cz puller under the effect of a TMF,Cz Si growth under a TMF (contd),FEMAGSoft 2013,Hartmann boundary layers along the melt-crystal and melt-crucible interfaces and associated boundary layer meshes. Strong Hartmann backflows develop.,Growth of a 300 mm diameter Si
11、 crystal under the effect of a 0.5 T TMF.,Cz Si growth under a TMF (contd),FEMAGSoft 2013,Cz Si growth under a TMF (contd),Czochralski growth of a silicon crystal under a 500 mT horizontal magnetic field,Global simulation the growth of 300 mm and 400 mm crystals,FEMAGSoft 2013,Cz Si growth under a T
12、MF (contd),Growth of a 300 mm crystal under a 500 mT TMF,Left: melt surface Right: meridional cross-sections parallel and perpendicular to the magnetic field,Top: velocity field Bottom: temperature field,FEMAGSoft 2013,Cz Si growth under a TMF (contd),Growth of a 400 mm crystal under a 500 mT TMF,Le
13、ft: melt surface Right: meridional cross-sections parallel and perpendicular to the magnetic field,Top: velocity field Bottom: temperature field,FEMAGSoft 2013,Czochralski growth of a silicon crystal under a 3000 or 5000 G horizontal magnetic field,Global simulation of the growth of 300 mm and 400 m
14、m crystals,Cz Si growth under a TMF (contd),FEMAGSoft 2013,Geometry and operating conditions,Cz Si growth under a TMF (contd),FEMAGSoft 2013,Description of the simulations,Cz Si growth under a TMF (contd),FEMAGSoft 2013,Crystal diameter: 400 mm Pulling rate: 0.45 mm/min Crucible rotation rate: 5 RPM
15、 Magnetic field type: horizontal Magnetic field strength: 3000 G Gas flow: no,Temperature field (K): top left: parallel cross section; bottom left: perpendicular cross section; top right: top view,Case 1,Cz Si growth under a TMF (contd),FEMAGSoft 2013,Case 1,Crystal diameter: 400 mm Pulling rate: 0.
16、45 mm/min Crucible rotation rate: 5 RPM Magnetic field type: horizontal Magnetic field strength: 3000 G Gas flow: no,Velocity field (m.s-1): top left: parallel cross section; bottom left: perpendicular cross section,Cz Si growth under a TMF (contd),FEMAGSoft 2013,Case 1,Crystal diameter: 400 mm Pulling rate: 0.45 mm/min Crucible rotation rate: 5 RPM Magnetic field type: horizontal Magnetic field strength: 3000 G Gas flow: no,
