Measurement and Thermodynamic Analysis of Carbon Solubility in Si–Cr Alloys at SiC Saturation

Abstract

Si–Cr alloy is one of the predominant solvents for rapid solution growth of 4H–SiC crystals. The solubilities of carbon in Si–40 mol%Cr alloy at SiC saturation at 1773–2273 K and in Si–Cr alloys of various chromium contents at 2073 K were measured by equilibrating the Si–Cr alloy with a 4H–SiC single crystal. Carbon solubility in Si–40 mol%Cr alloy increased with temperature from 0.22 mol% at 1773 K to 3.59 mol% at 2273 K. At 2073 K, carbon solubility at SiC saturation increased with the chromium content in the liquid from 0.18 mol% in Si–20 mol%Cr to 16.4 mol% in Si–80 mol%Cr. A thermodynamic analysis of the Si–Cr–C alloy was also conducted. Although the sub-regular solution model is often adopted to estimate phase relations in solution systems, this predicted a carbon solubility in Si-40 mol%Cr at SiC saturation more than two times higher than the measured value. In contrast, a quasi-chemical model that considered the competition between substitutional Si and Cr atoms bonding to interstitial carbon atoms reproduced the activity coefficient of carbon in Si–Cr alloys of 60–100 mol%Si composition, in which the carbon solubility at SiC saturation was less than 1.5 mol%, fairly well. This quasi-chemical model enabled the precise phase relation to be evaluated when designing the solution growth of SiC using a Si–Cr solvent.

1. Introduction

Power devices using silicon carbide (SiC) are expected to be useful for various applications, especially those in the mid-voltage range (between 500 and 1500 V). 4H–SiC bulk crystals fabricated by physical vapor transport have been used as the substrates for such power devices; however, the high dislocation densities of SiC substrates fabricated by this technique, ranging from 10² to 10⁴ cm⁻², are reported to cause device defects¹⁾.

Solution growth has attracted great attention for the production of 4H–SiC crystals with low dislocation density because the growth conditions are almost at thermal equilibrium. Recently, the growth of crystals with ultra-low dislocation density (< 0.1/cm²) and production of dislocation-free crystals by top-seeded solution growth have been reported²⁾. Both inadequate growth rates and solvent inclusions in the grown crystal caused by interfacial roughening are important issues in the production of bulk SiC crystal. To maintain a smooth interface under rapid growth, it is important to achieve high carbon solubility and suitable supersaturation by incorporating additives into the Si-based solvent^3–6). For example, rapid growth (exceeding 2 mm/h) while maintaining a smooth interface has been achieved using Si–40 mol%Cr-based solvent⁷⁾, although supersaturation was not specifically discussed. Accurate thermodynamic information on liquid Si-based alloys is essential if a high growth rate with a smooth interface is to be achieved.

The estimation of carbon solubility in alloy solvents has previously been conducted by employing the sub-regular solution model for the liquid phase using the thermodynamic properties of binary⁸⁾ or binary and ternary⁹⁾ alloys; however, carbon solubility in Si–X binary solutions has only been measured for a limited number of systems^8,10) and applicability of the sub-regular solution model to the estimation of carbon solubility in Si-based systems has not yet been fully verified. Si–Cr-based alloys are one of the predominant solvents for the growth of SiC, so precise data on carbon solubility for Si–Cr alloys are important for optimum control of the growth process.

In the present study, we measured the carbon solubility in Si–40 mol%Cr alloy at SiC saturation at 1773–2273 K and that in Si–Cr alloys with various chromium contents at 2073 K. Thereafter, thermodynamic analysis was carried out to determine a suitable model for describing the thermodynamic properties of Si–Cr–C solutions.

2. Experimental Procedure

The equilibrium experiments were carried out using an induction furnace equipped with a quartz reaction tube, as shown in Fig. 1. Silicon (0.05–0.10 g) (11 N purity) and chromium (99.9% purity) were weighed to give compositions of Si–20 mol%Cr to Si–80 mol%Cr and placed on a 4H–SiC single crystal in a graphite crucible. The samples were heated to 1773–2273 K in an Ar–10%H₂ ambient atmosphere. The temperature of the undersurface of a SiC polycrystal placed just below the 4H–SiC single crystal was measured using a mono-colored pyrometer, which was calibrated using the melting points of Si, Fe, Pt, and Al₂O₃. After holding for 10 min, which was predetermined from preliminary experiments, a sample droplet was sucked up using a tantalum capillary. Sample that adhered to the tantalum was then subjected to the combustion–infrared absorption method to determine its carbon concentration. From the measured mass fraction of carbon in the collected alloy, the mole fraction of carbon and silicon at SiC saturation are given by eqs. (1)–(3):   

\[ X_{i} = (W_{i}/M_{i})/ (W_{\rm C}/M_{\rm C} + W_{\rm Si}/M_{\rm Si} + W_{\rm Cr}/M_{\rm Cr}); \](1)

  

\[ W_{\rm Cr} = \{1 - (M_{\rm C} + M_{\rm Si})/M_{\rm C} \times W_{\rm C}\} \times W_{\rm Cr}^{0}; \](2)

  

\[ \begin{split} W_{\rm Si} = & \{1 - (M_{\rm C} + M_{\rm Si})/M_{\rm C} \times W_{\rm C}\}\\ & \times W_{\rm Si}^{0} + M_{\rm Si}/M_{\rm C} \times W_{\rm C}, \end{split} \](3)

where M_i [g], W_i, and X_i are the atomic weight, mass fraction, and mole fraction of component i in the collected alloy, respectively. $W_{i}^{0}$ is the mass fraction of i in the initial alloy.

Fig. 1

Schematic illustration of experimental apparatus for measurement of carbon solubility in Si–Cr alloy.

3. Results and Discussion

3.1 Carbon solubility in the Si–Cr system at SiC saturation

The top view of a droplet heated to 2073 K before being collected is shown in Fig. 2. No parasitic crystals, which would have been generated by an uneven temperature distribution in the droplet, were observed. The measured solubilities are summarized alongside the experimental conditions in Table 1.

Fig. 2

Top view of Si–Cr droplet on 4H–SiC substrate at 2073 K.

Table 1 Experimental results.

No.	Initial composition	Temperature, T/K	Mass%C	Mol%C	Mol%Si
1	Si-40 mol%Cr	1773	0.149	0.46	59.91
2	Si-40 mol%Cr	1773	0.073	0.23	59.95
3	Si-40 mol%Cr	1773	0.072	0.22	59.96
4	Si-40 mol%Cr	1873	0.109	0.34	59.93
5	Si-40 mol%Cr	1923	0.192	0.60	59.88
6	Si-40 mol%Cr	1923	0.231	0.72	59.86
7	Si-40 mol%Cr	1923	0.233	0.73	59.85
8	Si-40 mol%Cr	1923	0.210	0.65	59.87
9	Si-40 mol%Cr	1973	0.391	1.21	59.76
10	Si-40 mol%Cr	1973	0.311	0.97	59.81
11	Si-40 mol%Cr	1973	0.220	0.69	59.86
12	Si-40 mol%Cr	2073	0.442	1.37	59.73
13	Si-40 mol%Cr	2073	0.424	1.31	59.74
14	Si-40 mol%Cr	2073	0.350	1.09	59.78
15	Si-40 mol%Cr	2073	0.472	1.46	59.71
16	Si-40 mol%Cr	2173	0.670	2.06	59.59
17	Si-40 mol%Cr	2173	0.767	2.35	59.53
18	Si-40 mol%Cr	2173	0.713	2.19	59.56
19	Si-40 mol%Cr	2273	1.05	3.19	59.36
20	Si-40 mol%Cr	2273	1.19	3.59	59.28
21	Si-40 mol%Cr	2273	1.06	3.21	59.36
22	Si-20 mol%Cr	2073	0.067	0.18	79.89
23	Si-30 mol%Cr	2073	0.119	0.35	69.86
24	Si-50 mol%Cr	2073	1.08	3.47	50.00
25	Si-60 mol%Cr	2073	2.13	6.96	41.39
26	Si-70 mol%Cr	2073	3.70	11.99	34.79
27	Si-80 mol%Cr	2073	5.14	16.40	29.84

The logarithm of the measured carbon solubility in the Si–40 mol%Cr alloy saturated with SiC at 1773–2273 K is plotted with closed circles in Fig. 3. The carbon solubility at SiC saturation increased with temperature from 0.22 mol% at 1773 K to 3.59 mol% at 2273 K. Moreover, the logarithm of carbon solubility was almost directly proportional to the reciprocal temperature. This indicates that carbon dissolution obeys the Van't Hoff equation according to the standard Gibbs energy of SiC formation (eq. (5)) where Henry's law is established:   

\[ \begin{split} & {\rm Si} ({\rm l}) + {\rm C} ({\rm l}) \to {\rm SiC} ({\rm s})\\ &\quad \Delta G^{\circ} = - 233\ 100 + 61.01T({\rm J/mol}) \end{split} \](4)^11,12)

  

\[ \Delta G^{\circ} = - RT{\rm ln} \frac{a_{\rm SiC(s)}}{a_{\rm Si(l)}\cdot a_{\rm C(l)}} = RT{\rm ln}(a_{\rm Si(l)} \cdot \gamma_{\rm C(l)} \cdot X_{\rm C}), \](5)

where a_i and γ_i are the activity and activity coefficient of component i, respectively, and a_i(s) and a_i(l) are the activity of i at the standard states of pure solid and pure liquid, respectively.

Fig. 3

Measured carbon solubility in Si–40 ml%Cr alloy saturated with SiC.

Figure 4 shows the composition dependence of the logarithm of carbon solubility at SiC saturation measured at 2073 K, with the solubility obtained when pure chromium was reacted with SiC¹³⁾. The carbon solubility increased with increasing chromium content in the liquid, from 0.18 mol% in Si–20 mol%Cr alloy to 16.4 mol% in Si–80 mol%Cr alloy.

Fig. 4

Measured carbon solubility in Si–20 mol%Cr to Si–80 mol%Cr alloys saturated with SiC at 2073 K.

3.2 Thermodynamic analysis using sub-regular solution model

To estimate the carbon solubility in solutions used for the solution growth of SiC, researchers have previously employed the sub-regular solution model for the liquid phase using regular solution parameters of the constituent binary systems. In this section, a comparison is made between the measured carbon solubility and its value estimated by the sub-regular solution model, as in previous work⁸⁾. The thermodynamic properties used^10,14,15) are listed in Table 2. Note that similar liquidus compositions at SiC saturation were confirmed using a different dataset⁹⁾.

Table 2 Thermodynamic properties of the liquid phase in C–Cr, Cr–Si, and C–Si systems.

System	Sub-regular solution parameter, $ L_{\rm i - j}^{n} $/J mol−1	Researcher
C-Cr	$ L_{\rm C - Cr}^{0} = - 127,957 - 7.6695T $ $ L_{\rm C - Cr}^{1} = 79,574 $ $ L_{\rm C - Cr}^{2} = 86,315 $	Teng et al.14)
Cr-Si	$ L_{\rm Cr - Si}^{0} = - 119,216.57 + 16.11445T $ $ L_{\rm Cr - Si}^1 = - 47,614.70 + 12.17363T $	Coughanowr et al.15)
C-Si	$ L_{\rm C - Si}^{0} = 8,700 $	Kawanishi et al.10)

The estimated carbon solubility in the Si–Cr alloy at SiC saturation is plotted with broken lines in Figs. 3 and 4. In Fig. 3, the sub-regular solution model overestimated the solubility by more than a factor of two greater than the measured value over the entire temperature range. The estimated composition dependence of the carbon solubility at SiC saturation, shown in Fig. 4, was higher than the measured value over the entire composition range, and especially in the X_Si range from 0.4 to 0.6. The fraction of Si–Cr bonds is its largest at such intermediate compositions in the binary Si–Cr system. When carbon dissolves in Si–Cr alloy, it disturbs the attractive interaction between silicon and chromium and, in return, is subjected to a repulsive interaction, which results in a low carbon solubility. The sub-regular solution partly considers such interactions in the alloy, but does not adequately reproduce the carbon solubility. Hence, it is inferred that the sub-regular solution model is not suitable for estimating the thermodynamic properties of Si–Cr–C alloy.

3.3 Thermodynamic analysis using quasi-chemical model

We assumed that the inconsistency between the measured carbon solubilities at SiC saturation and those estimated using the sub-regular solution model is attributable to the behavior of carbon in the liquid Si–Cr alloy. In the sub-regular solution model, the components are considered to share the substitutional sites; however, the atomic radius of carbon is much smaller than those of silicon and chromium¹⁶⁾. We therefore presumed that carbon does not behave substitutionally, but interstitially, in Si–Cr alloy. Here, we adopted the quasi-chemical model formulated by Jacob and Alcock¹⁷⁾, in which competition between the bonding of interstitial (solute) atoms and substitutional (solvent) atoms is considered. The assumptions of this model are: (1) the solute interstitial carbon obeys Henry's law; (2) the solute atoms bond to solvent atoms with coordination number n; (3) solvent atoms are bonded to each other with coordination number Z when solute carbon is not dissolved and bonded with coordination number Z^* when they have direct interaction with the solute. The fractional change in the coordination number of the solvent atoms caused by the incorporation of solute atoms, t, can be expressed as:   

\[ t = \frac{Z - Z^{*}}{Z}, \](6)

where t is a parameter, the value of which can vary with change of the solvent composition. Subsequently, we obtained the following expression for the activity coefficient of carbon in Si–Cr alloys¹⁷⁾:   

\[ \frac{1}{[\gamma_{\rm C}^{0}]^{1/n}} = X_{\rm Si} \left\{ \frac{{\gamma_{\rm Si}}^{t}}{[\gamma_{\rm C(Si)}^0]^{1/n}} \right\} + X_{\rm Cr} \left\{ \frac{{\gamma_{\rm Cr}}^{t}}{[\gamma_{\rm C(Cr)}^0]^{1/n}} \right\}, \](7)

where $\gamma_{{\rm C}(j)}^{\circ}$ is the activity coefficient of carbon at infinite dilution in pure j (Si or Cr), and γ_Si and γ_Cr for the Si–Cr alloy were obtained from the Gibbs energy of the Si–Cr alloy¹⁷⁾. Otsuka¹⁸⁾ suggested that either n = 4, t = 1/2¹⁷⁾ or n = 6, t = 1/3¹⁹⁾ are indispensable within a useful range of accuracy, which implies the condition of n × t = 2. This model was proposed for systems in which solute–solvent bonds are much stronger than solvent–solvent bonds.

The activity coefficient was also derived from the measured carbon solubility at SiC saturation (Section 3.1) by considering the equilibrium between SiC and the alloy. Equation (5) holds when the alloy is saturated with SiC. The same thermodynamic value¹⁵⁾ was used for the calculation of activity of silicon in Si–Cr alloys; that at 2073 K is shown in Fig. 5. Here, we assumed that the activity of silicon in the SiC-saturated liquid Si–Cr–C alloy corresponded to that in the binary Si–Cr alloy at the same silicon concentration. In Fig. 6, the activity coefficients derived from the experimental carbon solubilities at SiC saturation are compared with the activity coefficients estimated by eq. (7) and by the sub-regular solution model. The value estimated by the sub-regular solution model greatly deviates from the measured value over the entire composition range. The value estimated by the quasi-chemical model (for both n = 4, t = 1/2 and n = 6, t = 1/3) agrees well with the measured value for X_Si (≧ 0.6). The estimated curve deviated from the activity coefficients derived from measured solubilities as X_Si decreased. Note that the experimental liquid composition toward high carbon and high chromium compositions at smaller X_Si deviated from the composition of the hypothesized binary Si–Cr alloy. This positive deviation from the estimated values at small X_Si is likely because the real a_Si of Si–Cr–C alloys should be larger than that in the hypothesized binary Si–Cr alloy, which might have resulted in an overestimation of the activity of the carbon for the experimental value.

Fig. 5

Estimated activity of silicon in Si–Cr alloys at 2073 K.

Fig. 6

Activity coefficient of carbon in Si–Cr alloys saturated with SiC at 2073 K from measured solubility and estimated from the quasi-chemical and sub-regular solution models.

Figure 7 shows the temperature dependence of the activity coefficient of carbon in Si–40 mol%Cr alloy. Similar to the behavior seen in Fig. 6, the values estimated by the sub-regular solution model significantly differ from the experimental values over the whole temperature range, while those estimated by the quasi-chemical model reproduce the measured values fairly well, especially in the lower temperature range. Regarding n and t, the estimation for n = 6, t = 1/3 had a more positive deviation than that for n = 4, t = 1/2, although there was no significant deviation between the two datasets. Taking these results into account, the present quasi-chemical model, which considers competition between substitutional silicon and chromium atoms bonding to interstitial carbon atoms, reproduces the thermodynamic properties of carbon in Si–Cr alloy well at less than 1.5 mol% carbon content. For higher carbon contents, a model applicable to concentrated solutions of carbon should be proposed.

Fig. 7

Activity coefficient of carbon in Si–40 mol%Cr alloy from measured solubility and estimated from quasi-chemical and sub-regular solution models.

4. Conclusions

The solubilities of carbon in Si–40 mol%Cr alloy saturated with SiC at 1773–2273 K and in Si–Cr alloy with various chromium contents at 2073 K were measured, and thermodynamic analysis was performed to reproduce the measured solubilities. The following results were obtained:

(1) Carbon solubility in Si–40 mol%Cr alloy at SiC saturation increased from 0.22 mol% at 1773 K to 3.59 mol% at 2273 K. When the chromium content of the liquid Si–Cr alloy was increased at 2073 K, the carbon solubility at SiC saturation increased from 0.18 mol% in Si–20 mol%Cr alloy to 16.4 mol% in Si–80 mol%Cr alloy.

(2) The sub-regular solution model overestimates carbon solubility in Si–Cr alloys at SiC saturation, especially at intermediate Si–Cr alloy compositions.

(3) The quasi-chemical model, which considers the competition between substitutional Si and Cr atoms bonding to interstitial carbon atoms, reproduces the activity coefficient of carbon well, especially at low carbon compositions (<1.5 mol%).

Acknowledgments

The authors are grateful to R. Takahashi of the University of Tokyo for her support during experiments. We also thank Kathryn Sole, PhD, from Edanz Group (www.edanzediting.com/ac) for editing a draft of this manuscript.

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