thermomechanical modeling of dislocation density increase … · 2015-11-20 · context ah model...

Thermomechanical modeling of dislocation density increase during PVT growth of SiC crystals

David Jauffrès Jean-Marc Dedulle Didier Chaussende Kanaparin Ariyawong

(present affiliation)

Context AH model Cooling simulation Prospects Conclusion

[email protected] COMSOL Conference Grenoble 2015 2

Context and motivation : Physical Vapor Transport and dislocations

Alexander-Haasen model & COMSOL implementation

Cooling of a mono-crystal and increase of dislocation density

Prospects : crystal growth and 3D modeling

Conclusion

Outline



Physical Vapor Transport growth of single SiC crystals

Applications : abrasives, wafers for power electronic devices …

Plasticity (T=2300 °C !)

Induction heating Very high temperatures (2300 °C)

Dislocation density increase

Thermal gradients

Thermal stresses

Context

(Wikipedia)

LMGP PVT reactor

(Wikipedia)



Mechanical behavior of SiC at very high temperature

Highly anisotropic viscoplasticity

Lara et al., Ceram. Int. 38, 1381–1390 (2012).

Single hcp crystal Dislocations glide along basal plane (0001)

AH model



Alexander-Haasen model : viscoplasticity with internal variable

−=

TkQbNm

B

neff

vp exp 0 τνγ

(mobile) dislocation density / m-2

Effective stress

Shear strain rate / s-1

NmDeff −=ττ

−=

∂∂ +

TkQNmK

tNm

B

neff exp0

λτν

5 parameters to be fitted

λν ; ; ; ; 0 KQn

AH model



Alexander-Haasen model : COMSOL implementation

Calibration compression test (3D)

Coordinate system associated with behavior law

NB : nonlinear structural materials module is required

AH model







−=

∂∂ +

TkQNmK

tNm

B

neff exp0

λτν

Domain ODE governing dislocation density rate

AH model







−=

∂∂ +

TkQNmK

tNm

B

neff exp0

λτν

Domain ODE governing dislocation density rate

Anisotropic elasticity + creep strain rate

−=

TkQbNm

B

neff

vp exp 0 τνγ

AH model



Alexander-Haasen model : calibration / validation



NB : nonlinear structural materials module is required Gao et al., Cryst. Growth Des., 14, 1272-1278 (2014)

Strain

Stre

ss /

Pa

1.1 ; 5-7E ; 3.3 ; 2.8 ; 15-8.5E 0 ===== λν KeVQn

AH model



Alexander-Haasen model : calibration / validation



NB : nonlinear structural materials module is required Gao et al., Cryst. Growth Des., 14, 1272-1278 (2014)

Strain

Stre

ss /

Pa

1.1 ; 5-7E ; 3.3 ; 2.8 ; 15-8.5E 0 ===== λν KeVQn

T = 2300 K

phi = pi/4

OK with Schmid law

AH model



Crystal shape and thermal field from thermo-chemical simulation[1]

Cooling

Time-dependant heat transfer simulation of the whole reactor +

Time-dependant mechanical simulation of thermal stresses with temperature dependant behavior for the crystal

[1] Ariyawong, K. et al., Materials Science Forum, 778-780, 35-38 (2014).

Cooling simulation



Effect cooling velocity

Cooling profiles

Induction heating = internal heat source

Slow cooling minimize axial temperature gradient

T drop / K

Axia

l T g

radi

ent /

K.m

-1

Dislocation density increase very low under 1800K

T drop / K

Dislo

catio

n de

nsity

m

ultip

licat

ion

rate

/ m

-2s-1

Cooling simulation



Induction heating = internal heat source

Cooling simulation

Effect cooling velocity

Cooling profiles



Prospects : dynamic crystal growth simulation

Motivation : dislocations actually mainly develop during growth !

Growth velocity Thermal field

Thermo-chemical simulation[1]

[1] Ariyawong, K. et al., Materials Science Forum, 778-780, 35-38 (2014).

Deformed mesh node with prescribed mesh velocity

+ Mechanical modeling (AH)

Prospects



Prospects : 3D modeling

Motivation : off-axis orientation of the crystal

3D thermal field

2D axisymetric heat transfer simulation of the whole reactor

3D mechanical model 3 basal slip systems α

-> 3 additional ODE for Nm(α)

Prospects

Cylindrical crystal with simple convection cooling on top

Dislocation density increase after 50s cooling / m-2



Prospects : 3D modeling

Motivation : off-axis orientation of the crystal

3D thermal field

2D axisymetric heat transfer simulation of the whole reactor

3D mechanical model 3 basal slip systems α

-> 3 additional ODE for Nm(α)

Prospects

Off-axis cylindrical crystal with simple convection cooling on top

Dislocation density increase after 50s cooling / m-2

8° disorientation



COMSOL implementation of AH model for crystal plasticity (basal slip systems) Additional domain ODE + user-defined creep law

Dislocation density is tracked with an internal variable Nm

Effect of cooling velocity can be studied

Two directions for future work : (1) simulation of dislocation increase during growth (2) 3D modeling for off-axis crystals

Conclusion


[email protected] COMSOL Conference Grenoble 2015

18



19

Extra slide : Schmid law

Gao et al., Cryst. Growth Des., 14, 1272-1278 (2014)

Strain

Stre

ss /

Pa

1.1 ; 5-7E ; 3.3 ; 2.8 ; 15-8.5E 0 ===== λν KeVQn

T = 2300 K

phi = pi/4 Schmid law λφστ coscos=

)2/cos()cos(21

4/ phiphiphiphi

−=

= πσσ

π



20

Extra slide : CRSS decrease with T

0

10

20

30

40

50

60

70

80

1000 1500 2000 2500

CR

SS [M

Pa]

Temperature [K]

Exp. Data @ 1.5e-4s-1 (Lara 2012)

exp(13656/T-6.688)

CRSS ~ 0.5 MPa @ 2300 K



21

Extra slide : mesh convergence



Cylindrical crystal

Radial cooling vs axial cooling

Axisymetric model ; imposed quadratic thermal fields



Cylindrical crystal

Effect of crystal height

Axisymetric model ; cylindric crystal ; imposed quadratic thermal fields

Diapositive numéro 1Diapositive numéro 2Diapositive numéro 3Diapositive numéro 4Diapositive numéro 5Diapositive numéro 6Diapositive numéro 7Diapositive numéro 8Diapositive numéro 9Diapositive numéro 10Diapositive numéro 11Diapositive numéro 12Diapositive numéro 13Diapositive numéro 14Diapositive numéro 15Diapositive numéro 16Diapositive numéro 17 Diapositive numéro 22Diapositive numéro 23

thermomechanical modeling of dislocation density increase … · 2015-11-20 · context ah model...

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