modeling and numerical simulation of the application of traveling … · crystal growth from the...
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Weierstrass Institute forApplied Analysis and Stochastics
Leibniz Institute for
Crystal Growth
Modeling and numerical simulation of the
application of traveling magnetic fields to stabilize
crystal growth from the melt
O. Klein (WIAS + MATHEON, Berlin)
joined work with:Ch. Grützmacher, P.-E.-Druet, J. Sprekels (WIAS + MATHEON, Berlin),
Ch. Lechner (TU Vienna), P. Philip (LMU, Munich),
Ch. Frank-Rotsch, F.-M. Kießling, W. Miller, U. Rehse, (IKZ, Berlin),
P. Rudolph (CTC, Schönefeld)Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de
12th International Conference on Free Boundary Problems 2012, June 11th-15th, 2012
Magnetic fields and crystal growth processes
• Semi-conductor mono-crystals are
important for high-technology devices
• Many of the crystals: grown from the melt
by growth processes of Czochralski type
• To improve the growth processes, one
needs to stabilizes the melt movement
• (Time-dependent) magnetic field can
stabilizes the melt movement
• Typically, the magnetic fields are
generated by induction coils placed
outside of the growth apparatus
• Growth of III–V compounds requires to
use pressure chambers with thick steel
walls walls diminish the magnetic field
generated by external coilsShow mpg video Show avi video
r[m]z[
m]
0 0.1 0.2 0.3
0.4
0.5
0.6
Im(Pot.) [Wb/m]: -0.003 -0.001 0.001 0.003
inductioncoils
?
6
resistanceheater
6
steel
crystal
?melt
producing a field of sufficient magnitude in
the melt requires much energy,
TMF stabilizing stabilize crystal growth from the melt FBP 2012, June 11th-15th 2 (13) R©
Project R© 07/2005 – 06/2008
• Cooperation of :
– Leibniz Institute of Crystal Growth (IKZ)
ETP (Hannover)
IISB (Erlangen)
– WIAS– Steremat Elektrowärme GmbH, Berlin– AUTEAM Industrie-Elektronik GmbH,
Brandenburg
• Internal heater-magnet modules (HMM), i.e.
coil–formed resistance heaters, and
electrical components to use them have
been developed
• Replacing the usual meander-formed
resistance heater units in the growth vessel
by an HMM
one can generate appropriate fields in the
melt with moderate power consumption
r[m]z[
m]
0.05 0.1 0.15 0.2
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Im(Pot.) [Wb/m]
0.00018E-056E-054E-052E-050
-2E-05-4E-05-6E-05-8E-05-0.0001
Coil 1120o
Coil 20o
Coil 3-120o
Show mpg video Show avi video
Innovation award Berlin-Brandenburg 2008
given to the project
,
TMF stabilizing stabilize crystal growth from the melt FBP 2012, June 11th-15th 3 (13) R©
Full Model of Magneto-Hydrodynamics
• Navier-Stokes equations in Boussinesq approximation: for melt in Ω1
ρ1 (∂u
∂t+ (u · ∇)u) = −∇p+ div(2 η(θ)Du) + f(θ) + j × µH,
div u = 0 in ]0, T [×Ω1 .
• Maxwell’s equations
j = curlH = 0 , div(σc E) = 0, in ]0, T [×Ωnc ,
j = curlH = σc(θ) (E + u× µH) in ]0, T [×Ωc
curlE + µ∂H
∂t= 0 , div(µH) = 0 , in ]0, T [×Ω .
• Energy balance
ρ1 c (∂θ
∂t+ u · ∇θ) = div(κ(θ)∇θ) + η(θ)D(u, u) +
|j|2
σc(θ)in ]0, T [×Ω1,
ρ c∂θ
∂t= div(κ(θ)∇θ) +
|j|2
σc(θ)in ]0, T [×Ωi (i 6= 1) .
,
TMF stabilizing stabilize crystal growth from the melt FBP 2012, June 11th-15th 4 (13) R©
Radiative heat transfer between surfaces of cavities:
− κgas(θgas)∇θgas • ~ngas −R+ J = −κ(θ)∇θ • ~ngas
• ~ngas: outer unit normal w.r.t. gas phase,
• total outgoing radiation R = σεT 4glo + (1− ε)J
– σ Boltzmann radiation constant, ε emissivity
• incoming radiation
J(x) =∫
ΓΛ(x, y)ω(x, y)R(y)dy
– Γ boundary of cavity– Λ(x, y) = 1 if y is “visible” from x, 0
otherwise– ω(x, y) view factors
ω(x, y) :=
(~ngas(y) • (x− y)
) (~ngas(x) • (y − x)
)π((y − x) • (y − x)
)2
,
TMF stabilizing stabilize crystal growth from the melt FBP 2012, June 11th-15th 5 (13) R©
Publications
• Concept of HMM: Rudolph JCG 2008
• Modeling and Simulation: Lechner–K.–Druet JCG 2007
K.–Lechner–Druet–Philip–Sprekels–Frank-Rotsch–Kießling–Miller–Rehse–Rudolph JCG
2008, MHD 2009
Rudolph–Czupalla–Dropka–Frank-Rotsch–Kiessling-K.-Lux-Miller-Rehse-Root JKCGC
2009
Dropka–Miller–Rehse–Rudolph–Buellesfeld–Sahr–K. –Reinhardt JCG 2011
Dreyer–Druet–K.–Sprekels WIAS 2012
• Existence of solutions: Druet Thesis 2009, MMAS 2009, CzMJ 2009, NA-RWA 2009, ApM
2010,
• Optimal control problem: Druet–K.–Sprekels–Tröltzsch–Yousept SIAM JCO 2011
• Free Boundary Problem: Druet WIAS 2011, WIAS 2012,
Contributed Talk Wed. 11.00
P.-É. Druet is researcher in the project C9 “Simulation and Optimization of Semiconductor
Crystal Growth from the Melt Controlled by Traveling Magnetic Fields” in the DFG-Research
Center FZT 86, MATHEON, (Heads: O. Kl., J. Sprekels, F. Tröltzsch)
,
TMF stabilizing stabilize crystal growth from the melt FBP 2012, June 11th-15th 6 (13) R©
Global simulations
• main parts of crystal growth device are simulated,
axially symmetric approximation of real geometry is
considered
• melt motion is ignored
• time-harmonic version Maxwell equation
• stationary heat equation
• Software WIAS-HiTNIHS (P. Philip, O. Kl.)
WIAS-High Temperature Numerical Induction
Heating Simulator (partially developed in MATHEON
Project C9)
• LPA Mark 3 in a configuration for LEC crystal
growth of GaAs
4 kg GaAs melt, diameter=15.2 cm, height=4.5 cm
GaAs melt covered by a boric oxide layer with a
height of 1.35 cm
Control T at triple point by adapting power used in
simulation
300
500
1000
1300
1600
300
temp. [K]
17001600150014001300120011001000900800700600500400300
Geometry Temperature
Pressurechamber
melt
Crystal
Insulation
Heater-magnetmodule (HMM)
,
TMF stabilizing stabilize crystal growth from the melt FBP 2012, June 11th-15th 7 (13) R©
Extended HMM
Special HMM following the
patent DE 10 2007 028 548
by Ch. Frank-Rotsch, P.
Rudolph, O. Kl., R.-P.
Lange, B. Nacke:• 3 coils surrounding the
crucible, each having 5
rings, producing a
downwards moving
TMF,
• 2 additional spirals
below the crucible,
producing an outward
moving TMF
1300
1500
1250
1450
1550
1700
1650
temp. [K]170016501600155015001450140013501300125012001150110010501000
Geometry Temperature
melt
crystal
extendedHeater-magnetmodule (HMM)
?
Show mpg video Show avi video
,
TMF stabilizing stabilize crystal growth from the melt FBP 2012, June 11th-15th 8 (13) R©
Global simulation 7→ local simulation
• time average over one period of electromagnetic fields, computed form time-harmonic
representation 7→ Lorentz-force density for local simulation
• Lorentz force density in melt generated by the HMM in the LPA Mark 3:
r
z
0.02 0.04 0.06
0.37
0.38
0.39
0.4
0.41
500 N/m^3
crystalboricoxide
• Temperature and heat fluxes in the melt computed by global simulations are also used as
initial and boundary data for local simulation in the melt
,
TMF stabilizing stabilize crystal growth from the melt FBP 2012, June 11th-15th 9 (13) R©
Local simulation (Ch. Lechner using NAVIER)
• only domain filled with melt is considered.
• heat sources by induction current ignored
• motion induced current ~u× ~B = ~u× µ ~H neglected Maxwell equation are
decoupled from melt motion
• Implementation by Ch. Lechner in the framework of NAVIER (E. Bänsch)
Snapshots of computed velocity and temperature distribution
without Lorentz forceΤ [Κ]
15241522152015181516151415121510150815061504150215001498
3.5 cm/sCrystal
Frame 001 6 Oct 2009 mho_tnbc_v22211_5rpm_0lor_1
Crystal
Frame 002 6 Oct 2009 mho_tnbc_v22211_5rpm_0lor_1
maximal velocity≈ 3.5 cms
with Lorentz forceΤ [Κ]
15181516151415121510150815061504150215001498
8.5 cm/sCrystal
Frame 001 6 Oct 2009 mho_tnbc_v22211_5rpm_lor_2
Crystal
Frame 002 6 Oct 2009 mho_tnbc_v22211_5rpm_lor_2
maximal velocity≈ 8.5 cms
,
TMF stabilizing stabilize crystal growth from the melt FBP 2012, June 11th-15th 10 (13) R©
Temperature oscillations in a monitor point (Ch. Lechner using NAVIER)
1508
1510
1512
1514
1516
1518
1520
1522
320 340 360 380 400 420 440 460
T [K
]
t [s]
without Lorentz force
main temperature oscillations≈ 8K
1508
1510
1512
1514
1516
1518
1520
1522
320 340 360 380 400 420 440 460
T [K
]
t [s]
with Lorentz force
main temperature oscillations≈ 2K
frequency increased
,
TMF stabilizing stabilize crystal growth from the melt FBP 2012, June 11th-15th 11 (13) R©
Project R©/ Project AVANTSOLAR (2008-2011)
• Conclusion project R©:
– Numerical simulations + crystal growth experiments show
Lorentz forces generated by an internal HMM can influence the melt flow
during crystal growth
using an extended HMM and appropriate TMFs, we can improve the growth
conditions
– an extended HMM has successfully been used for LEC crystal growth at the IKZ
• Project AVANTSOLAR (2008-2011):
–R©project partners + SCHOTT Solar Wafer GmbH + two other
research institutes– using traveling magnetic fields generated by an internal HMM to improve directional
solidification of solar-grad silicon– successfully growth of 640 kg Si ingots
,
TMF stabilizing stabilize crystal growth from the melt FBP 2012, June 11th-15th 12 (13) R©
Funding
• Projects R©(07/2005–06/2008) and AVANTSOLAR (07/2008–06/2011)
were supported by
– the German Federal State of Berlin in the framework of the “Zukunftsfonds Berlin”,– the Technology Foundation Innovation Center Berlin (TSB),– cofinanced by the European Union within the European Regional Development
Fund (EFRE). Investing in your future.
• Project C9 “Simulation and Optimization of Semiconductor Crystal Growth from the Melt
Controlled by Traveling Magnetic Fields” (05/2002–05/2014) is supported by:
,
TMF stabilizing stabilize crystal growth from the melt FBP 2012, June 11th-15th 13 (13) R©