Effects of the Temperature Gradient on the Fracture Stress of Large-Sized SiC Grown by PVT Method

doi:10.16553/j.cnki.issn1000-985x.2025.0109

Abstract

Abstract: Fracture stress remains the primary barrier preventing the diameter of silicon carbide （SiC） single crystals grown by the physical vapor transport （PVT） method from exceeding 200 mm. In the present study， the fracture stress was calculated under both 4° off-axis and on-axis growth conditions. The results demonstrate comparable fracture behavior between the two growth conditions， with negligible contributions from basal plane slips and nearly identical effects of prismatic plane slips. Besides， the effects of the temperature gradient on the fracture stress were elaborated， suggesting that almost all fracture stresses arise from the radial temperature gradient at high temperatures， while the axial temperature gradient exhibits minimal effect. Further simulations investigated the effects of temporal shape evolution， crystal convexity， and diameter， revealing a consistent correlation between fracture stress magnitude and radial temperature gradient variation. This study provides furthsinsights into the fracture stress-temperature gradient relationship， offering guidance for fracture prevention during PVT growth.

Key words: silicon carbide; physical vapor transport （PVT） method; numerical simulation; single crystal growth; fracture stress; temperature gradient

CLC Number:

TB34

XU Binjie, CHEN Pengyang, LU Sheng’ou, XUAN Lingling, WANG Anqi, WANG Fan, PI Xiaodong, YANG Deren, HAN Xuefeng. Effects of the Temperature Gradient on the Fracture Stress of Large-Sized SiC Grown by PVT Method[J]. Journal of Synthetic Crystals, 2025, 54(12): 2083-2100.

Figures/Tables 21

Fig.1 （a） Schematic of the core section for the 3D global model of the crucible；（b） mesh structure for the 3D thermal stress calculation；（c） schematic diagram of the transformation from 2D temperature distribution to 3D temperature distribution

Fig.2 Left half： physical model of SiC PVT growth； right half： meshing of the computational model， with the enlarged view showing the dynamic mesh boundary conditions conditions， here， Pi(s)* and Pi(p)* represent the saturated equilibrium pressures of each gas-phase component at the seed crystal and powder source surfaces， respectively， where i denotes the gas-phase components Si， Si2C， and SiC2

Fig.3 （a） Hexagonal unit cell of SiC with schematic diagram of basal （blue） and prismatic （yellow） plane slips， where a represents the unit vectors of the slip direction of BPDs， b and c represent the unit normal vectors of the prismatic plane and basal planes， respectively；（b） basal plane of the hexagonal crystal system of SiC

Table 1 Unit vectors of the slip direction and the unit normal vectors of the slip plane for different slip systems under on-axis and 4° off-axis growth conditions in the Cartesian coordinate system

Vector	On-axis	4° off-axis
$c$	$0 0 1$	$- s i n 4 ° 0 c o s 4 °$
$a 1$	$12 - 32 0$	$12 c o s 4 ° - 32 12 s i n 4 °$
$a 2$	$12 32 0$	$12 c o s 4 ° 32 12 s i n 4 °$
$a 3$	$- 1 0 0$	$- c o s 4 ° 0 - s i n 4 °$
$b 1$	$0 1 0$	$0 1 0$
$b 2$	$- 32 - 12 0$	$- 32 c o s 4 ° - 12 - 32 s i n 4 °$
$b 3$	$32 - 12 0$	$32 c o s 4 ° - 12 32 s i n 4 °$

Vector	On-axis	4° off-axis
$c$	$0 0 1$	$- s i n 4 ° 0 c o s 4 °$
$a 1$	$12 - 32 0$	$12 c o s 4 ° - 32 12 s i n 4 °$
$a 2$	$12 32 0$	$12 c o s 4 ° 32 12 s i n 4 °$
$a 3$	$- 1 0 0$	$- c o s 4 ° 0 - s i n 4 °$
$b 1$	$0 1 0$	$0 1 0$
$b 2$	$- 32 - 12 0$	$- 32 c o s 4 ° - 12 - 32 s i n 4 °$
$b 3$	$32 - 12 0$	$32 c o s 4 ° - 12 32 s i n 4 °$

Table 2 Slip systems of basal and prismatic plane slips in the Cartesian coordinate system under on-axis growth

Slip system		$s$	$n$
	$α 1$	$- 1 0 0$	$0 0 1$
Basal plane	$α 2$	$12 - 32 0$	$0 0 1$
	$α 3$	$12 32 0$	$0 0 1$
	$α 1$	$- 1 0 0$	$0 1 0$
Prismatic plane	$α 2$	$12 - 32 0$	$- 32 - 12 0$
	$α 3$	$12 32 0$	$32 - 12 0$

Table 2 Slip systems of basal and prismatic plane slips in the Cartesian coordinate system under on-axis growth

Slip system		$s$	$n$
	$α 1$	$- 1 0 0$	$0 0 1$
Basal plane	$α 2$	$12 - 32 0$	$0 0 1$
	$α 3$	$12 32 0$	$0 0 1$
	$α 1$	$- 1 0 0$	$0 1 0$
Prismatic plane	$α 2$	$12 - 32 0$	$- 32 - 12 0$
	$α 3$	$12 32 0$	$32 - 12 0$

Table 3 Linear relationship between RSS and the stress tensors in the Cartesian coordinate system for basal plane and prismatic plane slips under on-axis growth

Basal plane	$τ x z$	$τ y z$
$τ R S S α 1$	$- 1$
$τ R S S α 2$	$12$	$- 32$
$τ R S S α 3$	$12$	$32$
Prismatic plane	$σ x x$	$σ y y$	$τ x y$
$τ R S S α 1$			$- 1$
$τ R S S α 2$	$- 34$	$34$	$12$
$τ R S S α 3$	$34$	$- 34$	$12$

Table 3 Linear relationship between RSS and the stress tensors in the Cartesian coordinate system for basal plane and prismatic plane slips under on-axis growth

Basal plane	$τ x z$	$τ y z$
$τ R S S α 1$	$- 1$
$τ R S S α 2$	$12$	$- 32$
$τ R S S α 3$	$12$	$32$
Prismatic plane	$σ x x$	$σ y y$	$τ x y$
$τ R S S α 1$			$- 1$
$τ R S S α 2$	$- 34$	$34$	$12$
$τ R S S α 3$	$34$	$- 34$	$12$

Table 4 Linear relationship between the plastic strain tensors in the Cartesian coordinate system and the plastic strains in each slip system for basal and prismatic plane slips under on-axis growth

Basal plane	$ε α 1$	$ε α 2$	$ε α 3$
$γ x z p$	$- 1$	$12$	$12$
$γ y z p$		$- 32$	$32$
Prismatic plane	$ε α 1$	$ε α 2$	$ε α 3$
$ε x x p$		$- 34$	$34$
$ε y y p$		$34$	$- 34$
$γ x y p$	$- 1$	$12$	$12$

Basal plane	$ε α 1$	$ε α 2$	$ε α 3$
$γ x z p$	$- 1$	$12$	$12$
$γ y z p$		$- 32$	$32$
Prismatic plane	$ε α 1$	$ε α 2$	$ε α 3$
$ε x x p$		$- 34$	$34$
$ε y y p$		$34$	$- 34$
$γ x y p$	$- 1$	$12$	$12$

Fig.4 Schematic diagrams of a 4H-SiC crystal and wafer under off-axis grown. （a） Side view of the SiC crystal；（b） top view of the SiC wafer；（c） side view of the SiC wafer；（d） crystal coordinate system coordinate system （colored） with an off-angle and the Cartesian coordinate system （dashed line）；（e） cylindrical coordinate with an off-angle

Table 5 Slip systems of basal and prismatic plane slips in the Cartesian coordinate system under 4° off-axis growth

Slip System		$s$	$n$
	$α 1$	$- c o s 4 ° 0 - s i n 4 °$	$- s i n 4 ° 0 c o s 4 °$
Basal plane	$α 2$	$12 c o s 4 ° - 32 12 s i n 4 °$	$- s i n 4 ° 0 c o s 4 °$
	$α 3$	$12 c o s 4 ° 32 12 s i n 4 °$	$- s i n 4 ° 0 c o s 4 °$
	$α 1$	$- c o s 4 ° 0 - s i n 4 °$	$0 1 0$
Prismatic plane	$α 2$	$12 c o s 4 ° - 32 12 s i n 4 °$	$- 32 c o s 4 ° - 12 - 32 s i n 4 °$
	$α 3$	$12 c o s 4 ° 32 12 s i n 4 °$	$32 c o s 4 ° - 12 32 s i n 4 °$

Table 5 Slip systems of basal and prismatic plane slips in the Cartesian coordinate system under 4° off-axis growth

Slip System		$s$	$n$
	$α 1$	$- c o s 4 ° 0 - s i n 4 °$	$- s i n 4 ° 0 c o s 4 °$
Basal plane	$α 2$	$12 c o s 4 ° - 32 12 s i n 4 °$	$- s i n 4 ° 0 c o s 4 °$
	$α 3$	$12 c o s 4 ° 32 12 s i n 4 °$	$- s i n 4 ° 0 c o s 4 °$
	$α 1$	$- c o s 4 ° 0 - s i n 4 °$	$0 1 0$
Prismatic plane	$α 2$	$12 c o s 4 ° - 32 12 s i n 4 °$	$- 32 c o s 4 ° - 12 - 32 s i n 4 °$
	$α 3$	$12 c o s 4 ° 32 12 s i n 4 °$	$32 c o s 4 ° - 12 32 s i n 4 °$

Table 6 Linear relationship between RSS and the stress tensors in the Cartesian coordinate system for basal plane and prismatic plane slips under 4° off-axis growth

Basal plane	$σ x x$	$σ z z$	$τ x y$	$τ x z$	$τ y z$
$τ R S S α 1$	$12 s i n 8 °$	$- 12 s i n 8 °$		$- c o s 8 °$
$τ R S S α 2$	$- 14 s i n 8 °$	$14 s i n 8 °$	$32 s i n 4 °$	$12 c o s 8 °$	$- 32 c o s 4 °$
$τ R S S α 3$	$- 14 s i n 8 °$	$14 s i n 8 °$	$- 32 s i n 4 °$	$12 c o s 8 °$	$32 c o s 4 °$
Prismatic plane	$σ x x$	$σ y y$	$σ z z$	$τ x y$	$τ x z$	$τ y z$
$τ R S S α 1$				$- c o s 4 °$		$- s i n 4 °$
$τ R S S α 2$	$- 34 c o s 2 4 °$	$34$	$- 34 s i n 2 4 °$	$12 c o s 4 °$	$- 34 s i n 8 °$	$12 s i n 4 °$
$τ R S S α 3$	$34 c o s 2 4 °$	$- 34$	$34 s i n 2 4 °$	$12 c o s 4 °$	$34 s i n 8 °$	$12 s i n 4 °$

Table 6 Linear relationship between RSS and the stress tensors in the Cartesian coordinate system for basal plane and prismatic plane slips under 4° off-axis growth

Basal plane	$σ x x$	$σ z z$	$τ x y$	$τ x z$	$τ y z$
$τ R S S α 1$	$12 s i n 8 °$	$- 12 s i n 8 °$		$- c o s 8 °$
$τ R S S α 2$	$- 14 s i n 8 °$	$14 s i n 8 °$	$32 s i n 4 °$	$12 c o s 8 °$	$- 32 c o s 4 °$
$τ R S S α 3$	$- 14 s i n 8 °$	$14 s i n 8 °$	$- 32 s i n 4 °$	$12 c o s 8 °$	$32 c o s 4 °$
Prismatic plane	$σ x x$	$σ y y$	$σ z z$	$τ x y$	$τ x z$	$τ y z$
$τ R S S α 1$				$- c o s 4 °$		$- s i n 4 °$
$τ R S S α 2$	$- 34 c o s 2 4 °$	$34$	$- 34 s i n 2 4 °$	$12 c o s 4 °$	$- 34 s i n 8 °$	$12 s i n 4 °$
$τ R S S α 3$	$34 c o s 2 4 °$	$- 34$	$34 s i n 2 4 °$	$12 c o s 4 °$	$34 s i n 8 °$	$12 s i n 4 °$

Table 7 Linear relationship between the plastic strain tensors in the Cartesian coordinate system and the plastic strains in each slip system for basal and prismatic plane slips under 4° off-axis growth

Basal plane	$ε α 1$	$ε α 2$	$ε α 3$
$ε x x p$	$12 s i n 8 °$	$- 14 s i n 8 °$	$- 14 s i n 8 °$
$ε z z p$	$- 12 s i n 8 °$	$14 s i n 8 °$	$14 s i n 8 °$
$γ x y p$		$32 s i n 4 °$	$- 32 s i n 4 °$
$γ x z p$	$- c o s 8 °$	$12 c o s 8 °$	$12 c o s 8 °$
$γ y z p$		$- 32 c o s 4 °$	$32 c o s 4 °$
Prismatic plane	$ε α 1$	$ε α 2$	$ε α 3$
$ε x x p$		$- 34 c o s 2 4 °$	$34 c o s 2 4 °$
$ε y y p$		$34$	$- 34$
$ε z z p$		$- 34 s i n 2 4 °$	$34 s i n 2 4 °$
$γ x y p$	$- c o s 4 °$	$12 c o s 4 °$	$12 c o s 4 °$
$γ x z p$		$- 34 s i n 8 °$	$34 s i n 8 °$
$γ y z p$	$- s i n 4 °$	$12 s i n 4 °$	$12 s i n 4 °$

Basal plane	$ε α 1$	$ε α 2$	$ε α 3$
$ε x x p$	$12 s i n 8 °$	$- 14 s i n 8 °$	$- 14 s i n 8 °$
$ε z z p$	$- 12 s i n 8 °$	$14 s i n 8 °$	$14 s i n 8 °$
$γ x y p$		$32 s i n 4 °$	$- 32 s i n 4 °$
$γ x z p$	$- c o s 8 °$	$12 c o s 8 °$	$12 c o s 8 °$
$γ y z p$		$- 32 c o s 4 °$	$32 c o s 4 °$
Prismatic plane	$ε α 1$	$ε α 2$	$ε α 3$
$ε x x p$		$- 34 c o s 2 4 °$	$34 c o s 2 4 °$
$ε y y p$		$34$	$- 34$
$ε z z p$		$- 34 s i n 2 4 °$	$34 s i n 2 4 °$
$γ x y p$	$- c o s 4 °$	$12 c o s 4 °$	$12 c o s 4 °$
$γ x z p$		$- 34 s i n 8 °$	$34 s i n 8 °$
$γ y z p$	$- s i n 4 °$	$12 s i n 4 °$	$12 s i n 4 °$

Fig.5 Temperature profile and distribution of the elastic component of σφφ （σφφe） of SiC crystal under on-axis and 4° off-axis growth for cooling time of 0 and 480 min

Fig.6 （a）~（c） Half-sectional 3D view of the RSS distributions in the basal plane slip systems α1， α2， and α3 under on-axis growth， with section planes taken as ［11ˉ00］，［01ˉ10］，［101ˉ0］ crystallographic planes， respectively. The non-displayed half-sections are symmetrically distributed with respect to the displayed half-sections along their respective section planes；（d）~（f） 3D view of the RSS distributions in the basal plane slip systems α1， α2， and α3， respectively， under 4° off-axis growth；（g）~（i） 3D view of the RSS distributions in the prismatic plane slip systems α1， α2， and α3， respectively， under on-axis growth， exhibiting perfect 120° rotational symmetry among the 3 systems；（j）~（l） 3D view of the RSS distributions in the prismatic plane slip systems α1， α2， and α3， respectively， under 4° off-axis growth， maintaining approximate 120° rotational symmetry among the 3 systems

Fig.7 （a）~（c） Half-sectional 3D view of the plastic strain distributions caused by the basal plane slip systems α1， α2， and α3 under on-axis growth， with section planes taken as ［11ˉ00］，［01ˉ10］，［101ˉ0］ crystallographic planes， respectively， the non-displayed half-sections are symmetrically distributed with respect to the displayed half-sections along their respective section planes；（d）~（f） 3D view of the plastic strain distributions caused by the basal plane slip systems α1， α2， and α3， respectively， under 4° off-axis growth；（g）~（i） 3D view of the plastic strain distributions caused by the prismatic plane slip systems α1， α2， and α3， respectively， under on-axis growth， exhibiting perfect 120° rotational symmetry among the 3 systems；（j）~（l） 3D view of the plastic strain distributions caused by the prismatic plane slip systems α1， α2， and α3， respectively， under 4° off-axis growth， maintaining approximate 120° rotational symmetry among the 3 systems

Fig.8 3D view of the σφφ distribution after 480 min of cooling， accounting for the plastic strains due to basal plane slips （a），（b）， and prismatic plane slips （c），（d）. Where （a） and （c） are calculated under on-axis growth， and （b） and （d） are calculated under 4° off-axis growth

Fig.9 Diagrams of radial temperature distribution for the same radius （a） and different radiuses （b）

Fig.10 （a） Schematic diagram illustrating the variation in convexity and the radial temperature distribution；（b） radial temperature distribution curves on the seed surface for different convexities；（c） radial temperature distribution curves normalized by subtracting the values at the 100 mm radius （T100） for easier visual comparison；（d） evolution of the maximum RSS in the prismatic plane slip systems with cooling time for different convexities；（e） evolution of σφφe at the periphery of the seed surface with cooling time for different convexities， the inset shows the point where the σφφe value is extracted；（f） maximum RSS during high temperature growth for different convexities， calculated by considering axial strain （3D） and neglecting axial strain （2D-plane）， along with the ratio of the two；（g）~（i） distributions of 12σrr-σφφ， which represent the radial cross-section of the maximum RSS value at high temperature， for convexities of 0， 8， and 16 mm， respectively

Fig.11 （a） Schematic diagram illustrating the variation in diameter and the radial temperature distribution， with the crystal convexity maintained to be 0；（b） radial temperature distribution curves on the seed surface for different diameters；（c） maximum RSS during high temperature growth for different diameters， calculated by considering axial strain （3D） and neglecting axial strain （2D-plane）， along with the ratio of the two

Fig.12 （a） Evolution of the maximum RSS in the prismatic plane slip systems with cooling time for different diameters；（b） evolution of σφφe at the periphery of the seed surface with cooling time for different diameters， the inset shows an enlargement of the data points after 400 min of cooling

Fig.13 Evolution of the crystal interface， crystal meshing， and 12σrr-σφφ， which represent the radial cross-section of the maximum RSS in the prismatic plane slip systems， during crystal growth

Fig.14 （a） Radial temperature distribution curves on the seed surface for different growth time；（b） evolution of the maximum RSS during growth time， calculated by considering axial strain （3D） and neglecting axial strain （2D-plane）， along with the ratio of the two；（c） evolution of σφφe at the periphery of the seed surface during growth time

References 48

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