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©