Optimization of power control in the reduction of basal plane dislocations during PVT growth of 4H-SiC single crystals

Journal of Crystal Growth 392 (2014) 92–97

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Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro

Optimization of power control in the reduction of basal plane dislocations during PVT growth of 4H-SiC single crystals B. Gao n, K. Kakimoto Research Institute for Applied Mechanics, Kyushu University, Japan

art ic l e i nf o

a b s t r a c t

Article history: Received 20 December 2013 Received in revised form 30 January 2014 Accepted 3 February 2014 Communicated by: J.J. Derby Available online 12 February 2014

The influence of power control on the multiplication of basal plane dislocations (BPDs) during PVT growth of 4H-SiC single crystals was studied by numerical modeling. Three sets of different power histories during growth were tested: continuously increasing power, continuously decreasing power, and constant power. The results show that optimization of the power history control is crucial for the reduction of basal plane dislocations during growth. If only low BPD density is concerned, then constant low power is the best choice. However, if both low BPD density and high growth rate are desirable, then concave continuously increasing power is the best choice. & 2014 Elsevier B.V. All rights reserved.

Keywords: A1. Line defects A1. Computer simulation A2. Growth from vapor A2. Industrial crystallization

1. Introduction Wide-bandgap silicon carbide (SiC) is a promising semiconductor material for electronic and opto-electronic devices involving high power, high temperature, high frequency, and intense radiation owing to its stable chemical and thermal properties [1,2]. Bulk SiC crystals are commonly grown by the physical vapor transport (PVT) method [3,4]. In bulk SiC crystal growth, significant progress has been made in reducing the most damaging defects: micropipes. In 2007, researchers at Cree Inc. reduced the micropipe density (MPD) by 90% in 150-mm substrates [5]. As the density of micropipes in SiC crystals has been suppressed to a technologically tolerable level, the focus of quality improvement has shifted to less severely damaging defects such as dislocations [5]. There are several types of dislocations, including basal plane dislocations (BPDs), which are deformation-induced dislocations, and grown-in dislocations (threading edge dislocations (TEDs) and threading screw dislocations (TSDs)). Deformation-induced dislocations usually lie on the primary slip plane, which is the (0001) basal plane. Because BPDs have been associated with an increase in the number of defects observed in epitaxy and as the root cause of gate voltage drift in bipolar devices [6], reduction of the BPD density in SiC crystals has become a major focus in future quality improvement efforts [5]. BPDs are mainly generated and multiply during high-temperature processes, such as the crystal growth process and cooling processes. Thus, optimization of crystal growth and cooling processes could

n

Corresponding author. Tel.: þ 81 92 583 7744; fax: þ81 92 583 7743. E-mail address: [email protected] (B. Gao).

http://dx.doi.org/10.1016/j.jcrysgro.2014.02.005 0022-0248 & 2014 Elsevier B.V. All rights reserved.

reduce the generation of BPDs. However, study of the optimization process requires a good model that can correctly connect the generation of BPDs to the practical operational conditions, and also correctly describe the rate-dependent plastic deformation process of the SiC crystal. Gao et al. [7] extended the Alexander–Haasen model [8,9] to SiC crystals by including different activation enthalpies in different temperature regions. Their model showed good agreement between numerical and experimental data for stress–strain curves in a wide temperature range [7]. In this paper, the model proposed in Ref. 7 is extended to a threedimensional (3D) model, and the deformation-induced dislocations (BPDs) are considered to be responsible for the high-temperature plastic deformation [10–12]. Furthermore, resolved shear stress along the primary slip directions is used as the driving force for BPD multiplication. The improved model is used to study the influence of power control on the multiplication of BPDs during crystal growth of 4H-SiC. We aim to clarify the following problems: What type of power control during growth is best for reducing BPDs? What type of power control during growth is a good choice for reducing BPDs and also for increasing the growth rate? 2. Numerical simulations 2.1. Modeling of basal plane dislocations of 4H-SiC single crystal Hexagonal 4H-SiC crystals have three primary slip directions in the (0001) basal plane, which are defined as ½1120, ½1120, and ½1120 in the hexagonal coordinate system. The resolved shear stress in each slip direction can be calculated by the tensor

B. Gao, K. Kakimoto / Journal of Crystal Growth 392 (2014) 92–97

2.2. Design of power history during growth

transformation technique using stress components obtained from a 3D analysis [13] or an axisymmetric analysis [14,15]. Details of the basic procedure can be found in Ref. [13]. After the resolved shear stress is obtained, the basal plane dislocations can be calculated from the following equations [16]:

dt

ðαÞ ðαÞ ¼ Nm v b;

In the growth experiments, furnace is usually controlled by temperature profiles at two monitoring points, which is located at the top and the bottom of growth chamber. In this paper, in order to study the influence of power history on BPD evolution, the furnace is designed to be controlled by power profile. It may not be as sufficient as the temperature control procedure in many experiments. However, it can provide valuable guidance for growing high-quality SiC single crystal by controlling power history. To study the influence of power history on BPD evolution during growth, a series of numerical simulations with different histories of input power were performed. Each numerical simulation is given a case number from 1 to 8. Three sets of power histories were chosen and are shown in Fig. 1. For all of the numerical experiments, the power change started at t¼2.5 h and ended at t¼12.5 h. Before the power change, the crystal was allowed to grow at a constant power of 8398 W for cases 1, 3, 4, and 5, and 6743 W for cases 2, 6, 7, and 8. The three sets of power histories are summarized as follows:

ð1Þ

ð αÞ dN m ðαÞ ðαÞ ¼ K τðefαfÞλ Nm v b; dt

ð2Þ

where the subscript m denotes the mobile dislocation, b is a Burgers vector, N is the dislocation density, k is the multiplication constant, λ is the stress exponential factor, and εpl is the creep strain. τef f and v are the effective stress for dislocation multiplication and the slip velocity of dislocation, respectively, and they are defined by [17] ðαÞ τefðαfÞ ¼ τresolv  τdðαÞ ;

ð3Þ

  Q ðαÞp ; exp  vðαÞ ¼ k0 τef f kb T

ð4Þ

(a) The total power generated inside the graphite crucible was kept constant as shown in Fig. 1(a). There are two cases (cases 1 and 2) in this set. The two cases served as references for the cases with continuous power change described below. (b) The total power generated inside the graphite crucible was continuously decreased from 8398 W to 6743 W as shown in Fig. 1(b). There are three cases (cases 3, 4, and 5) in this set. Cases 3 and 5 are reflection symmetry about the center point of case 4. In case 3, due to a convex design of curve, the power first slowly decreases and then rapidly decreases; while in case 5, due to a concave design of curve, the power first rapidly decreases and then slowly decreases. Both cases 3 and 5 contain slowly and rapidly changing power regions, and the only difference between cases 3 and 5 is that the slowly or rapidly changing power regions are at different temperature levels. (c) The total power generated inside the graphite crucible was continuously increased from 6743 W to 8398 W. There are three cases (cases 6, 7, and 8) in this set. Cases 6 and 8 are reflection symmetry about the center point of case 7. The power curve is convex in case 6 and concave in case 8. For convenient comparison, the slowly or rapidly changing region in case 6 is identical to that in case 3, which means that cases 3, 5, 6, and 8 are coincidental after some rotation and translation.

ðαÞ where τresolv and τdðαÞ are the resolved shear stress and back-stress in α slip direction, respectively. k0 is a material constant, p is the stress exponential factor, Q is the activation enthalpy, kb is the Boltzmann constant, and T is temperature in Kelvin. After the dislocation densities and creep strains are calculated for all of the slip directions, the total dislocation density N m and total creep strain tensor εpl can be expressed as [14,15]: 3

ðαÞ ; Nm ¼ ∑ Nm

ð5Þ

α¼1 3

1 2

ðαÞ εpl ¼ ∑ εplðαÞ ðsðαÞ  nðαÞ þ nðαÞ  sðαÞ Þsignðτresolv Þ;

α¼1

ð6Þ

where s is the unit vector in slip direction, and n is the unit normal vector of slip plane. Table 1 gives all the parameters for the calculations. Table 1 Parameters for the dislocation calculations. Symbol

Description

Value or/and units

b k k0 Q

Burgers vector Multiplication constant Material constant [7] Activation enthalpy [7]

λ

Stress exponential factor [7]

p kb μ

Stress exponential factor [7] Boltzmann constant Shear modulus

3.073  10  10 (m) 7.0  10  5 8.5  10  15 3.3 eV at T 41000 1C 2.6 eV at T o 1000 1C 1.1 at T 41000 1C 0.6 at T o 1000 1C 2.8 8.615  10  5 (eV/K) 7.992  1010 (Pa)

8000 7500 7000 2

6500

2

4

6

8

Numerical optimization of power control was performed in a traditional SiC PVT furnace, which consists of induction coils, SiC power, graphite crucible, pedestal, insulation shield, reflectors and furnace walls [18]. The SiC powder is placed inside the crucible, and the seed is attached to the bottom of the lid of the crucible

8500

8500

1

Power in graphite (W)

Power in graphite (W)

8500

2.3. Furnace setup and heat and mass transfer inside furnace

10

Growth time (h)

12

Power in graphite (W)

ð αÞ dεpl

3

8000

4

7500 5

7000 6500

93

2

4

6

8

10

Growth time (h)

12

8000

6

7500

7

7000

8

6500

2

4

6

8

10

12

Growth time (h)

Fig. 1. Power histories for three sets of numerical experiments: (a) constant power, (b) continuously decreasing power, and (c) continuously increasing power.

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[18]. The diameter of seed is 50 mm. The furnace is assumed to be axisymmetric. An extra annular-square chamber is connected to the top of the main growth chamber by a small gap [19]. The gap can be adjusted from 0.1 mm to 3.0 mm. The growth chamber has a shape of frustum, which has a top diameter of 50 mm, a bottom diameter of 100 mm, and a height of 43 mm [19]. The detailed configurations of furnace can be referred to Refs. [18,19]. Based on the above experimental furnace, numerical simulation of crystal growth was performed. The crystal growth is associated with the calculation of growth rate. The growth rate was obtained by fully coupled and multi-species compressible flow solver, which considers compressible effect, convection effect, buoyancy effect, flow coupling between argon gas and species, and Stefan effect. This solver does not assume a perfect stoichiometric incorporation of Si and C atoms into the growing crystal as performed elsewhere [20–22] and also does not assume thermodynamic equilibrium chemical reactions at crystal and source surfaces as carried out elsewhere [20,23]. The evaporation and deposition flux (growth rate) is automatically provided according to the supersaturation of species at the seed or supercooling at the powder source. By using the above solver, a typical temperature field inside furnace for constant low power control (case 2) is shown in Fig. 2 (a) and the temperature and flow field inside growth chamber is shown in Fig. 2(b). The total gas pressure inside furnace is 1.5 Torr. The top and bottom temperature at the central axial line of growth chamber is 2342 K and 2392 K. Thus, the temperature difference between source and seed surfaces is 50 K.

2.4. BPD density distribution for different power control methods 2.4.1. Constant power control The distribution of BPD density inside the crystal for constant power control (cases 1 and 2) is shown in Fig. 3. The growth direction is from the top to the bottom. The cone-shape crystal

T (K) 2422 2280 2138 1995 1853 1711

Nm (cm-3) 4.5E+03 3.1E+03 2.1E+03 1.4E+03 9.8E+02 6.7E+02 4.6E+02 3.1E+02 2.1E+02 1.5E+02 1.0E+02

Fig. 3. Distribution of BPD density inside the crystal for constant power: (a) case 1 and (b) case 2.

originates from the frustum shape of growth chamber. The maximum BPD density inside crystal for case 1 is 4300 cm  2, while for case 2 it is only 972 cm  2. The only difference between the two cases is that case 2 has a lower power level than case 1. Therefore, a low power level is beneficial for growing crystals with low BPD density. Although a low power level is beneficial for growing crystals with low BPD density, the growth rate is limited. Since the growth time is the same in the two cases, the growth rate can be approximately measured by the thickness of the grown crystal at the axial line. The thickness is 9.1 mm for the high-power case and 4.1 mm for the low-power case. Thus, low power has a much slower growth rate than high power. Since a low power level has a low BPD density but a slow growth rate, and a high power level has a fast growth rate but a high BPD density, a natural question is: Can we design a power control scheme to obtain both a fast growth rate and a low BPD density? To answer the above question, we studied continuously increasing power (CIP) and continuously decreasing power (CDP). By comparing the BPD density distribution for different power control schemes, we aim to determine if it is possible to design a power control method that gives both high growth rate and low BPD density.

1569 1427 1284 1142 1000

T (K) 2399 2388 2378 2367 2357 2346

Fig. 2. (a) Temperature distribution inside furnace, (b) Temperature and flow field distribution inside growth chamber.

2.4.2. Continuously decreasing power (CDP) control The distributions of BPD density inside the crystal for CDP control (cases 3, 4 and 5) are shown in Fig. 4. The convex power control (case 3) is shown in Fig. 4(ai), the linear control (case 4) is shown in Fig. 4(bi), and the concave control (case 5) is shown in Fig. 4(ci), where the subscript i denotes the growth time. The BPD density for convex CDP is always higher than those for linear and concave CDP at any given growth time, and the BPD density for the concave CDP curve is lowest. For example, at the end of growth time (t ¼12.5 h), the maximum BPD density is 4727 cm  2 for convex CDP control, 1632 cm  2 for linear CDP control, and 1135 cm  2 for concave CDP control. The BPD density for concave CDP control is almost four times less than that for convex CDP control. Thus, for CDP control it is best to choose concave power control for effective reduction of the generation of BPDs. After analyzing the BPD density, the growth rate was analyzed by the thickness of the crystal at the axial line. The growth

B. Gao, K. Kakimoto / Journal of Crystal Growth 392 (2014) 92–97

t1=5.0 h

t2=7.5 h

t3=10 h

95

t4=12.5 h Nm (cm-3) 4.5E+03 3.1E+03 2.1E+03 1.4E+03 9.8E+02 6.7E+02 4.6E+02 3.1E+02 2.1E+02 1.5E+02 1.0E+02

Fig. 4. Distribution of BPD density inside the crystal for continuously decreasing power: (ai) case 3, (bi) case 4, and (ci) case 5.

thickness was 7.9 mm for the convex CDP case, 7.2 mm for the linear CDP case, and 7.0 mm for the concave CDP case. Interestingly, the growth thickness in the concave CDP case can reach 89% of the thickness in the convex CDP case. In a word, the concave CDP control can grow a crystal height with 89% of that grown in convex CDP control, but produces the BPD density with only 24% of that in the latter; Furthermore, the concave case consumes less energy than the convex case. Therefore, for CDP control, concave power control is the best choice for both a high growth rate and a low BPD generation. 2.4.3. Continuously increasing power (CIP) control The distributions of BPD density inside the crystal for CIP control (cases 6, 7 and 8) are shown in Fig. 5. The convex CIP control (case 6) is shown in Fig. 5(ai), the linear CIP control (case 7) is shown in Fig. 5(bi), and the concave CIP control (case 8) is shown in Fig. 5(ci), where the subscript i denotes the growth time. The maximum BPD density is 2000 cm  2 for convex CIP control, 2400 cm  2 for linear CIP control, and 564 cm  2 for concave CIP control. The maximum BPD density for convex CIP control is slightly less than that for linear CIP control, but the distribution area of high BPD density for convex CIP control is much greater than that for linear control. The BPD density for concave CIP control is lowest. Therefore, for CIP control it is best to choose concave control for effective reduction of BPD generation. However, concave CIP power control has the slowest growth rate. The growth thicknesses at the axial line for convex, linear, and concave CIP control are 7.8, 7.5, and 6.6 mm, respectively. Thus, the growth thickness for concave CIP control can reach 85% of the growth thickness for convex CIP control. In a word, the concave CIP control can grow a crystal height with 85% of that grown in convex CIP control, but produces the BPD density with only 28% of that in the latter; Furthermore, the concave CIP case consumes less energy than the convex CIP case. Therefore, for CIP control, concave power control is also the best choice for both high growth rate and low BPD generation. 2.5. Comparison between different power controls 2.5.1. Comparison of continuously decreasing and continuously increasing powers For continuously changing power, whatever CDP and CIP control, the concave power control is always the best choice for growing crystals with low BPD density and high growth rate. However, to find the optimum power control scheme, we need to

know which one is better because these two curves correspond to the same energy consumption during growth. The distributions of BPD density for cases 5 and 8 are plotted in Fig. 6 with the same contour range. It can be seen that concave CIP control (Fig. 6(b)) results in less BPD dislocations than CDP control. Therefore, to effectively reduce BPD dislocations it is better to use concave CIP control, rather than concave CDP control to grow SiC crystals. It is important to determine the reasons why CIP concave control is beneficial for growing low BPD SiC crystals. To investigate this, the temperature histories of two end points along the axial line (A and C) and two end points of the crystal side (B and D) are shown in Fig. 7. The temperature differences between these points gradually decreases for the CIP case, but gradually increases for the CDP case. For example, at the end of growth for the CDP case (t¼12.5 h), the temperature difference between points B and D is 9.5 K and the length between B and D is 7.5 mm, which corresponds to an average temperature gradient of 1.3 K/mm; the temperature difference between points C and D is 14.5 K and the length between C and D is 28.7 mm, which corresponds to an average temperature gradient of 0.51 K/mm. However, at the end of growth for the CIP case, the temperature difference between points B and D is 8.3 K and the length between B and D is 6.80 mm, which corresponds to an average temperature gradient of 1.2 K/mm; the temperature difference between points C and D is 13.2 K and the length between C and D is 28.30 mm, which corresponds to an average temperature gradient of 0.47 K/mm. Therefore, the temperature gradient in the CIP case is smaller than that in the CDP case, which might be the reason for the lower BPD density in concave CIP control than concave CDP control. Why does the temperature gradient inside the crystal become smaller in the CIP case? The temperature gradient inside the crystal is determined by the difference between heating flux from the graphite crucible and conduction flux inside the SiC crystal. When the temperature increases from 2350 to 2550 K, the conductivity of the graphite crucible decreases from 46.8 to 42.0 Wm  1K  1. However, the conductivity of the SiC crystal only decreases from 31.5 to 29.6 Wm  1K  1. Thus, the ratio of heating flux from the graphite crucible to the conduction flux inside the crystal is relatively decreased, which results in a decrease in the temperature gradient inside the crystal. As well as the temperature gradient, there might be other important reasons for lower BPD density in the CIP case than the CDP case. Fig. 7 shows that for the CIP case most of the growth time is located in the low-temperature region, which corresponds to relatively low mobility of BPDs.

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B. Gao, K. Kakimoto / Journal of Crystal Growth 392 (2014) 92–97

t1=5.0 h

t2=7.5 h

t3=10 h

t4=12.5 h Nm (cm -3)

2.6E+03 1.6E+03 9.8E+02 6.0E+02 3.7E+02 2.3E+02 1.4E+02 8.6E+01 5.3E+01 3.3E+01 2.0E+01 Fig. 5. Distribution of BPD density inside the crystal for continuously increasing power: (ai) case 6, (bi) case 7, and (ci) case 8.

Nm (cm-3) 4.5E+03 3.1E+03 2.1E+03 1.4E+03 9.8E+02 6.7E+02 4.6E+02 3.1E+02 2.1E+02 1.5E+02 1.0E+02 Fig. 6. Distribution of BPD density inside the crystal for (a) concave continuously decreasing power, and (b) concave continuously increasing power.

2650

B

A

2600

Temperature (K)

can be deduced that the fast-decreasing power or fast-increasing power at high temperature region is helpful to reduce temperature gradient inside crystal, which has been validated in the paper [24]. After comparing the BPD density of the two cases, the growth rates of the two cases were also compared. The thickness of the crystal at the axial line is 6.5 mm for concave CIP control and 7.0 mm for concave CDP control. The growth rate in the concave CIP case is quite close to that in the concave CDP case (the CIP growth rate is 93% of the CDP growth rate). Therefore, it is better to use concave CIP control because concave CIP control produces lower BPD density with a comparable growth rate to concave CDP control.

Continuously-decreased power

D

C

2550 B D A

2500

C

2450 Continuously-increased power

2400 D B

2350

C

A

4

6

8

10

12

Growth time (h) Fig. 7. Temperature histories of four points at the axial line and the side of the crystal for continuously decreasing and continuously increasing power.

Thus, CIP control produces lower BPD density than CDP control because CIP control has a smaller temperature gradient than CDP control owing to a rapid decrease of the conductivity in the graphite crucible compared with the SiC crystal. Furthermore, CIP control also has a lower growth temperature than CDP control for most of the growth time (Fig. 7). Whatever CDP control or CIP control, the concave power control always has a fast-changing power at high temperature region and a slowly changing power at low temperature region. Since at high-temperature region the mobility of BPDs is high, it

2.5.2. Comparison between constant power and continuously change power After finding an optimal power control scheme (concave CIP), it is necessary to determine whether concave CIP control is also superior to constant power control. Fig. 8 shows a comparison of the constant power control and concave CIP control. The constant high-power control case (case 1) is shown in Fig. 8(a), the constant low-power control case (case 2) is shown in Fig. 8(b), and the concave CIP control case (case 8) is shown in Fig. 8(c). The maximum BPD density is 4300 cm  2 for case 1, 260 cm  2 for case 2, and 584 cm  2 for case 8. Therefore, for constant power, only constant low power (case 2) results in SiC crystals with low BPD density. However, the low-power case always corresponds to a low growth rate. The thicknesses at the axial line for cases 1, 2, and 8 are 9.1, 4.1, and 6.5 mm, respectively. Therefore, if only BPD density is considered, then constant low-power control is the best choice, but if both BPD density and growth rate are considered, then concave CIP control is the best choice.

B. Gao, K. Kakimoto / Journal of Crystal Growth 392 (2014) 92–97

97

are considered, then concave continuously increasing power is the best choice.

Acknowledgments

Nm

(cm-3)

4.5E+03

This work was partly supported by the Japan Society for the Promotion of Science (Grant No. 24360012).

2.4E+03 1.3E+03

References

7.2E+02 3.9E+02 2.1E+02 1.2E+02 6.3E+01

[1] [2] [3] [4] [5]

3.4E+01 1.8E+01 1.0E+01

Fig. 8. Distribution of BPD density inside the crystal for (a) constant high-power control (case 1), (b) constant low-power control (case 2), and (c) concave continuously increasing power control.

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

3. Conclusions

[18]

The influence of power control on the multiplication of basal plane dislocations (BPDs) during PVT growth of 4H-SiC single crystals was studied by numerical modeling. Three sets of different power histories during growth were tested: continuously increasing power, continuously decreasing power, and constant power. The results show that optimization of power control is crucial for the reduction of basal plane dislocations during crystal growth. If only low BPD density is considered, then constant low power is the best choice, but if both low BPD density and high growth rate

[19] [20] [21] [22]

[23] [24]

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