Negligible Temperature Dependence of Nitrogen Solubility in Molten Silicon–Chromium Alloys at Middle Composition Range

Abstract

N solubility in molten Si-based alloys is an important property for controlling the concentration of N in SiC crystals, fabricated by the solution growth method. In this study, N solubility in Si–Cr alloys, which is the commonly used solvent for solution growth, was measured at 1953, 2023, and 2073 K, and then evaluated in terms of thermodynamics. The measured N solubilities in Si–40, 55, and 72.5 mol% Cr obeyed the Sieverts’ law, and increased with the increase in Cr concentration. For Si–40 and 55 mol% Cr alloys, the temperature dependence of the N solubility was negligibly small, or even slightly positive, which is opposite to those for pure Si and Cr. This reverse feature in the middle composition range was reproduced by the estimated activity coefficient of N by the quasi-chemical model, which assumes N as interstitial atoms in Si–Cr solvents. In addition, another feature was found for the estimated activity coefficient of N, that is, an upward convex against the Si–Cr composition. These characteristics arise from the negatively large heat of mixing of Si–Cr alloys and its relatively large temperature dependence.

1. Introduction

Nitrogen acts as a donor in SiC crystals by substituting C sites.^1–4) In the solution growth of SiC bulk single crystals, heavily N-doped n-type crystals with low resistivity have been fabricated using Si-based alloy solvents.^5–7) Because N incorporation into SiC occurs at the SiC/alloy interface, the thermodynamic property of N in Si-based alloys is a crucial factor in controlling the concentration of N in SiC. The gas/liquid equilibration of N is described as follows:   

\begin{equation} \frac{1}{2}\text{N$_{2}$(g)} = \underline{\text{N}}\ (\text{in melt}) \end{equation} (1)

  

\begin{equation} K_{(1)} = \frac{a_{\text{N}}}{\sqrt{P_{\text{N${_{2}}$}}}} = \frac{\gamma_{\text{N}}x_{\text{N}}}{\sqrt{P_{\text{N${_{2}}$}}}}, \end{equation} (2)

where K₍₁₎ is the equilibrium constant of eq. (1), and a_N, γ_N, and x_N are the activity, activity coefficient, and mole fraction of N in the melt, respectively. $P_{\text{N}_{2}}$ is the partial pressure of N₂. If x_N is sufficiently low, then γ_N is a constant value of $\gamma _{\text{N}}^{\circ }$ ($ = \mathop{\lim}\limits_{x_{\text{N}} \to 0}\gamma _{\text{N}}$), and N solubility (x_N) is proportional to the square root of the partial pressure of N₂ ($P_{\text{N}_{2}}{}^{1/2}$) according to Sieverts’ law. Although x_N has not been measured, a linear relation between the N concentration in SiC and $P_{\text{N}_{2}}{}^{1/2}$ in the solution growth ambient has been reported.⁵⁾ This implies a correlation with the Sieverts’ law, while no reports on x_N in Si-based alloys are available. In addition, the effect of the negatively large heat of mixing of Si-based alloys^8–10) on the thermodynamic behavior of interstitial N atoms in the alloys has not been discussed. In the present study, the thermodynamic properties of N in Si–Cr alloys, which are frequently used as the solvent for SiC growth, were investigated. The N solubility in Si–Cr alloys at 1953, 2023, and 2073 K was measured by the quenching method. Further, the activity coefficient of N in the alloys, which is inevitable for the prediction of N solubility in various $P_{\text{N}_{2}}$, temperatures, and alloy compositions, was thermodynamically evaluated.

2. Experimental Procedure

A schematic of the experimental apparatus is shown in Fig. 1. Alloys of Si–40, 55, and 72.5 mol%Cr were prepared by melting high purity Si lumps (> 7N) and Cr flakes (> 4N) in an arc furnace. Here, the selection of the alloy compositions comes from the typical solvents for the SiC growth.^6,7,11,12) After crashing the alloys, each alloy (0.5 g) was charged in an alumina crucible with an inner diameter of 5 mm, and the crucibles were set to the graphite holder. Using an induction furnace, the sample assembly was heated under evacuation (<0.01 Pa) until 1273 K. The Ar–H₂–N₂ gas, whose moisture and oxygen were removed by gas purification columns, was introduced into the reaction tube and continuously flowed at a rate of 120 cm³·min⁻¹ at 1 atm through an alumina tube to control $P_{\text{N}_{2}}$ to be 0.01–0.3 atm. To ensure a sufficient supply of gas to the alloy surface, the tube end was placed 10 mm above the hot graphite holder. Then the supplied gas reached the surface of the alloy after passing a hot zone of >10 mm in length. Thereafter, the sample was heated at a rate of 200 K·min⁻¹ to the experimental temperatures of 1953, 2023, and 2073 K. After holding for 10 min, which was preliminarily determined to reach N saturation, the power of the furnace was shut down and a cold graphite block was quickly touched to the top of the graphite holder for quenching. It should be noted that the graphite block was used to accelerate the solidification of the surface of the alloys, which would be effective to minimize desorption or adsorption of N at gas/melt interface during cooling. The entire alloy was charged into an oxygen/nitrogen analyzer (EMGA-620W, Horiba Ltd., Japan) by a combustion-thermal conductivity measurement method to measure the N concentration in the alloy.

Fig. 1

Schematic of experimental apparatus.

3. Results and Discussions

3.1 N solubility in molten Si–Cr alloys

The N solubility in Si–Cr alloys at 1953, 2023, and 2073 K is shown in Fig. 2 and listed in Table 1. For each alloy, the measured x_N linearly increased with $P_{\text{N}_{2}}{}^{1/2}$ at all temperatures, suggesting that the Si–Cr alloys obey the Sieverts’ law under the present conditions. In Si–40 mol%Cr, the N solubility was the lowest in the order of several hundreds of ppm, and increased with the increase in Cr concentration in the alloy, reaching the maximum content of 1.7 mol% in Si–72.5 mol%Cr alloy at 2023 K, which is the average value of the condition. The effect of temperature on x_N was insignificant for Si–40 and 55 mol%Cr alloys.

Fig. 2

Relationship between N solubility in Si–40, 55 and 72.5 mol% Cr alloys and ($P_{\text{N}_{2}}$)^1/2 under $P_{\text{N}_{2}}$ = 0.01–0.3 atm at 1953, 2023, and 2073 K.

Table 1 N solubility in Si–40, 55 and 72.5 mol% Cr alloys under P_N2 = 0.01–0.3 atm at 1953, 2023, and 2073 K.

When the standard of a_N is defined as a_N = 1 at $P_{\text{N}_{2}}=1$ atm, eq. (2) yields the following relation:   

\begin{equation} x_{\text{N}} = \frac{\sqrt{P_{\text{N${_{2}}$}}}}{\gamma_{\text{N}}} \end{equation} (3)

From eq. (3) and the gradient in the linear relations in Fig. 2, $\gamma _{\text{N}}^{\circ}$ was evaluated, as shown in Table 2, and its temperature dependence is shown in Fig. 3. The $\ln \gamma _{\text{N}}^{\circ }$ exhibits a linear relation with the reciprocal of temperature in each alloy, and decreases with an increase in the Cr concentration in the alloy. For each alloy, the following equations were obtained using the least squares method:   

\begin{align} \ln\gamma_{\text{N}}^{\circ} &= \frac{1.07 \times 10^{4}(\pm 0.04 \times 10^{4})}{T} + 1.53(\pm 0.21)\\ &\quad (\text{Si–40$\,$mol%Cr},\ 1953\unicode{x2013}2073\,\text{K}) \end{align} (4)

  

\begin{align} \ln\gamma_{\text{N}}^{\circ}& = \frac{2.57 \times 10^{3}(\pm 3.47 \times 10^{3})}{T} + 4.64(\pm 1.72)\\ &\quad (\text{Si–55$\,$mol%Cr},\ 1953\unicode{x2013}2073\,\text{K}) \end{align} (5)

  

\begin{align} \ln\gamma_{\text{N}}^{\circ} &= -\frac{7.24 \times 10^{3}}{T} + 7.04\\ &\quad (\text{Si–72.5$\,$mol%Cr},\ 2023\unicode{x2013}2073\,\text{K}) \end{align} (6)

where, T represents temperature in K. Here, the gradient of $\ln \gamma _{\text{N}}^{\circ}$ against T⁻¹ is slightly negative for both pure Si and Cr melts, which is provided by the following relations according to the reported Gibbs energy change in N dissolution (eq. (1)).^13,14)   

\begin{align} \ln\gamma_{\text{N(Si)}}^{\circ} &= -\frac{4.354 \times 10^{4}}{T} + 8.697\\ &\quad (\text{pure Si},\ 1693\unicode{x2013}1923\,\text{K})^{13)} \end{align} (7)

  

\begin{align} \ln\gamma_{\text{N(Cr)}}^{\circ} &= -\frac{6.904 \times 10^{4}}{T} + 5.073\\ &\quad (\text{pure Cr},\ 1873\unicode{x2013}2473\,\text{K})^{14)} \end{align} (8)

Note that large discrepancies exist in the reported Gibbs energy change depending on the researchers for both pure Si^13,15–18) and Cr,¹⁴⁾ and sometimes positive gradients have been suggested.^15,19) The main difficulty in evaluating Si is the sampling of the Si melt without any inclusions, as can be observed from the large scattering of the measured N concentration.^15,18) For pure Cr, determining N solubility is difficult because of the high volatility of Cr and the strong affinity for oxygen, which are the reasons for the error. Therefore, the most reliable values were carefully judged from the experimental conditions described in the literature, which are given as eqs. (7) and (8), and both exhibited a negative gradient. Among the Si–Cr alloys, only Si–72.5 mol%Cr exhibited a negative gradient of $\gamma _{\text{N}}^{\circ}$, while the gradients for Si–40 and 55 mol% Cr were slightly positive. This reverse phenomenon occurred because of the comparable x_N regardless of temperature in the middle composition range (see Fig. 2), and is discussed in the next section with thermodynamic modeling.

Table 2 Activity coefficient of N at infinite dilution ($\gamma _{\text{N}}^{\circ }$) in molten Si–40, 55 and 72.5 mol% Cr alloys.

Fig. 3

Temperature dependence of activity coefficient of N in molten Si–40, 55 and 72.5 mol% Cr alloys.

3.2 Thermodynamic evaluation of N in molten Si–Cr alloys

The thermodynamic expression of $\gamma _{\text{N}}^{\circ}$ in the binary alloy by a quasi-chemical model,²⁰⁾ which considers N as an interstitial solute element in the solvent Si–Cr alloy, is as follows:   

\begin{equation} \frac{1}{\gamma_{\text{N}}^{\circ}{}^{1/n}} = x_{\text{Si}}\left(\frac{\gamma_{\text{Si}}^{t}}{\gamma_{\text{N(Si)}}^{\circ}{}^{1/n}}\right) + x_{\text{Cr}}\left(\frac{\gamma_{\text{Cr}}^{t}}{\gamma_{\text{N(Cr)}}^{\circ}{}^{1/n}}\right) \end{equation} (9)

where x_i and γ_i (i = Si or Cr) are the mole fraction and activity coefficient of component i in Si–Cr alloy, respectively, and $\gamma _{\text{N}(i)}^{\circ}$ is the activity coefficient of N in pure i at infinite dilution. The constant values, n and t, represent the bond number of the solute (N atom) to solvent atoms (Si and Cr atoms), and the fraction of decrease in bond number between solvent atoms by the existence of coordinating N atoms, respectively. In the present study, a combination of values of n = 4 and t = 1/2, which was proposed by Jacob and Alcock,²⁰⁾ was utilized. The values of γ_i and $\gamma _{\text{N}(i)}^{\circ}$ were obtained using the thermodynamic data.^8,13,14) The compositional dependence of $\ln \gamma _{\text{N}}^{\circ}$ at 2073 K is shown in Fig. 4, together with those obtained from the present measurements and the literature.^13,14) The estimated $\ln \gamma _{\text{N}}^{\circ}$ slightly increased with the addition of Cr to Si at less than x_Cr ∼ 0.3, and then decreased. The experimentally obtained values agreed with the estimations, indicating that the behavior of N in molten Si–Cr alloy is well expressed by the quasi-chemical model. This upward convex of $\ln \gamma _{\text{N}}^{\circ}$ is similar to that in the case of solute C in molten Si–Cr, Si–Ni, and Si–Fe alloys.^21,22) Such repulsive interactions of solute N or C atoms against the solvents are owing to the strong affinity between the solvent atoms. In the case of Si-based alloys, the negatively large heat of mixing (H^ex) of Si-based alloys (e.g., H^ex is less than −30 kJ·mol⁻¹ at Si–(37 – 70) mol% Cr at 2073 K⁸⁾) contributed to the unique behaviors of interstitial atoms.

Fig. 4

Compositional dependence of activity coefficient of N in molten Si–Cr alloys at 2073 K. The values for pure Si and Cr were referred to the literature.^13,14) The estimated curve using the quasi-chemical model is also presented.

Figure 5 shows the contour lines of iso-$\ln \gamma _{\text{N}}^{\circ}$ in molten Si–Cr alloys at 1923–2323 K obtained by the quasi-chemical model together with the experimental results. The experimental values agreed well with the estimations, and the upward convex of $\ln \gamma _{\text{N}}^{\circ}$ against the composition was confirmed at all temperatures. In addition, the sign of the gradient of $\ln \gamma _{\text{N}}^{\circ}$ against reciprocal of temperature changed from minus to plus in the middle composition region, approximately Si–(30–60) mol% Cr. This is because of the relatively larger increase in γ_Si and γ_Cr with an increase in temperature, as indicated in eq. (9). Moreover, this opposite characteristic was experimentally observed for the Si–40 and 55 mol%Cr alloys (see Fig. 3 and eqs. (4)–(6)). Using Fig. 5 and eq. (3), the estimation of N solubility at various temperatures, compositions, and $P_{\text{N}_{2}}$ is available. As an example, contour lines of iso-N solubility at $P_{\text{N}_{2}}=0.1$ atm are shown in Fig. 6 together with the average of the experimental results. Note that the Si₃N₄ saturated region, appearing at high x_Si at <1961 K, is shown with a semi-transparent mask, and the estimation without considering Si₃N₄ formation is also presented to visualize the temperature dependence. The measured N solubilities were well reproduced by the quasi-chemical model, including the negligibly small, or even slightly increasing, trend toward temperature for Si–40 and 55 mol%Cr alloys. Therefore, the negligibly small temperature dependence of N solubility in molten Si–Cr alloys around the middle compositions was clarified by both experiments and thermodynamic estimations.

Fig. 5

Contour lines of iso-activity coefficient of N in molten Si–Cr alloys at 1923–2323 K obtained by the quasi-chemical model. The experimental results are also shown.

Fig. 6

Contour lines of iso-N solubility in molten Si–Cr alloys under $P_{\text{N}_{2}}=0.1$ atm at 1923–2323 K obtained by the quasi-chemical model. The Si₃N₄ saturated region is shown with a semi-transparent mask. The average results of experiments are also shown.

4. Conclusions

N solubility in molten Si–40, 55, and 72.5 mol% Cr alloys under $P_{\text{N}_{2}}=0.01-0.3$ atm at 1953, 2023, and 2073 K were measured and thermodynamically evaluated. The results are summarized as follows.

1. (1)    N solubility obeyed Sieverts’ law for each alloy at each temperature, and increased with the increase in Cr concentration in the alloy, with a maximum of 1.7 mol% in Si–72.5 mol% Cr at 2023 K.
2. (2)    Two unique N characteristics were found: (a) negligible temperature dependence of N solubility in the middle composition range and (b) upward convex of the activity coefficient of N against solvent composition. These features were explained by the large negative heat of mixing of the Si–Cr alloy and its relatively large temperature dependence.

Acknowledgments

The authors are grateful to Prof. Yoshikawa at the University of Tokyo for fruitful discussions on thermodynamics. The authors thank the IMRAM Central Analytical Facility for the use of the oxygen/nitrogen analyzer. This work was partly supported by JSPS KAKENHI (Grant Numbers 17H04960, 18K18934, and 19H00820) and the Cooperative Research Program of “Network Joint Research Center for Materials and Devices”.

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