New insight into the plastic deformation mechanisms during the SiO₂ phase transition process

Abstract

The removal of lattice impurities is the key to the purification of high-purity quartz (HPQ), especially for the intracell lattice impurities. Generally, the intracell lattice impurities can be migrated to the quartz surface via roasting, then removed by acid leaching. In order to reveal the phase transition of quartz during the roasting process, the evolution of structure, bond length, volume, lattice parameter and lattice stress in original, Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ substituted SiO₂ phases were employed to investigate the mechanisms of plastic deformation based on density functional theory calculations. Results showed that the evolution of bond lengths and volumes were mainly dominated by phase transition, and the interstitial volume in high temperature SiO₂ phases was higher than that in low temperature, indicating that the phase transition from α-quartz to β-cristobalite was beneficial to the migration of interstitial impurities. In addition, the phase transition from α-quartz to β-cristobalite needs to overcome the energy barriers while the phase transition from α-cristobalite to β-cristobalite needs to overcome the lattice stress. This study therefore provides an excellent theoretical basis for the plastic deformation mechanism, for the first time, beneficial to understanding the removal mechanisms of lattice impurities.

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1 Introduction

The quartz crucible prepared using high-purity quartz (HPQ) is a necessary container for monocrystalline silicon,^1,2 significantly influencing the healthy development of semiconductors, 5G communications, new energy and other industrial fields.³ The key to the purification of HPQ, especially for 5N HPQ (SiO₂ content ≥ 99.999%) is to remove the lattice impurities.^4–6 Normally, the lattice impurities can be divided into lattice substitution impurity and lattice interstitial impurities based on their location.^7,8 In terms of substitution configuration of a single Si atom, lattice impurities can be divided into three kinds, i.e., simple substitution (such as the substitution of Si⁴⁺ by Ti⁴⁺), substitution with interstitial charge compensator (such as the substitution of Si⁴⁺ by Al³⁺/Li⁺), and silanol groups (the substitution of Si⁴⁺ by 4H⁺).^9–13

Generally, the lattice impurities can be removed via roasting and leaching process.^10,14–16 Within which, the leaching process can only remove the surface impurities,¹⁰ while the lattice impurities can migrate from the intracell to surface during high-temperature roasting process. For instance, Wei et al.¹⁶ investigated the migration mechanisms during phase transition from α-quartz to β-quartz, suggesting that the lattice impurity of Al³⁺ and Li⁺ migrated from intracell to quartz sand surface via the c axis. Wu et al.¹⁷ indicated that the lattice impurities of Al³⁺, Fe³⁺, K⁺ and Ti⁴⁺ migrated from intracell to surface during the NH₄Cl roasting process.

In addition, quartz normally undergoes phase transition process during roasting. Götze et al.¹⁸ reported 15 polymorph SiO₂ phases, e.g., the most common phases were α-quartz, β-quartz, β-cristobalite and α-cristobalite, etc. The phase transition from α-quartz to β-quartz, from β-quartz to β-cristobalite and from β-cristobalite to α-cristobalite occurred at 573 °C, 1470 °C and 270 °C, respectively.^19–23 However, the plastic deformation mechanisms during these phase transitions were still unclear, which significantly affected the properties of SiO₂ lattice and the removement of lattice impurities. For instance, Wei et al.¹⁶ indicated that the interstitial volume was increased during the phase transition from α-quartz to β-quartz, beneficial to the migration of lattice interstitial impurities, such as Li⁺. Blankenburg et al.²⁴ found that the defects or twinning formed during the phase transition from α-quartz to β-quartz influenced the distribution of lattice impurities.

In this case, the phase transition of original and substitution SiO₂ phases were employed to investigate the plastic deformation mechanism during roasting process based on the density functional theory (DFT) calculation. Specifically, α-quartz, β-quartz, β-cristobalite and α-cristobalite were considered as the original SiO₂ phases, and Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ were considered as the impurity ion substituted in the crystal of quartz. This study, for the first time, clearly revealed the evolution of structure, bond length, volume, lattice parameter, lattice stress and energy in original, and substituted (Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺) SiO₂ phases, therefore providing an excellent theoretical basis for the plastic deformation during the phase transition process, beneficial to understanding the removal mechanisms of lattice impurities.

2 Methodology

Similar to our previous research,¹⁶ the generalized gradient approximation (GGA) with PW91 correlation potential was considered to perform the DFT calculation through Materials Studio in CASTEP package.²⁵ A kinetic energy cutoff of 360 eV was chosen for the geometric optimization calculation, and the Monkhorst–Pack scheme k-point meshes were 3 × 3 × 4, 2 × 2 × 3, 1 × 2 × 2 and 1 × 1 × 2 for α-quartz, β-quartz, β-cristobalite and α-cristobalite. The interstitial volume was calculated through the methods in our previous research.¹⁶ And the lattice stress was calculated via eqn (1):^26,27

S_L = −B × (ΔV/V) (1)

where S_L, B and V were the lattice stress, bulk modulus and bulk volume of SiO₂ phases, ΔV was the bulk volume changes between SiO₂ phases. The formation energy was calculated via eqn (2):²⁸

E_f = E_tol − aE_Si − bE_O − cE_Ti − dE_Al − eE_Li − fE_H (2)

where E_f was the formation energy, E_tol was the total energy of SiO₂ phase, a, b, c, d, e and f were the numbers of Si, O, Ti, Al, Li and H atoms in SiO₂ phase, respectively. E_Si, E_O, E_Ti, E_Al, E_Li and E_H were the energies of the constituents in their ambient-condition ground-state phases for Si, O, Ti, Al, Li and H atoms, respectively.

3 Results and discussion

3.1 Structure of original SiO₂ phases

In order to better compare the plastic deformation mechanisms between typical SiO₂ phases, the supercells with the same number of atoms (all in 72 atoms) of 2 × 2 × 2 α-quartz, 2 × 2 × 2 β-quartz, 3 × 1 × 1 β-cristobalite and 3 × 2 × 1 α-cristobalite were employed to perform the geometric optimization calculation, and the optimized structures were shown in Fig. 1.

Fig. 1 The structure of optimized typical SiO₂ phases (all in Si₂₄O₄₈).

These typical SiO₂ phases had similar arrangement of atoms, e.g., each Si atom was boned with four surrounding O atoms to form a silica tetrahedron (i.e., four Si–O bonds), and each O atom was boned with two surrounding Si atoms to connect silica tetrahedron.^16,29,30 However, the specific Si–O bond length and Si–O–Si angle were different for these typical SiO₂ phases, i.e., the Si–O bond length were 1.619/1.625 Å, 1.614 Å, 1.608 Å and 1.615/1.619 Å in α-quartz, β-quartz, β-cristobalite and α-cristobalite, while the Si–O–Si angle were 153.19^°, 144.47^°, 180.00^° and 153.27^°, respectively (Table 1), resulting in different crystal systems for these typical SiO₂ phases, i.e., trigonal, hexagonal, cubic and tetragonal for α-quartz, β-quartz, β-cristobalite and α-cristobalite, respectively.^18,20–23

Table 1 Lattice parameters of optimized typical SiO₂ phases

Lattice parameters	α-Quartz	β-Quartz	β-Cristobalite	α-Cristobalite
Supercell size	2 × 2 × 2	2 × 2 × 2	3 × 1 × 1	3 × 2 × 1
Atomic number	72	72	72	72
a (Å)	9.923	10.097	22.278	15.377
b (Å)	9.923	10.097	7.425	10.247
c (Å)	10.945	11.214	7.425	7.141
Volume (Å^3)	933.224	990.088	1228.300	1125.240
Si–O bond (Å)	1.619/1.625	1.614	1.608	1.615/1.619
O–Si–O angle (°)	153.19	144.47	180.00	153.27

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In addition, the bulk volumes were 933.224 ų, 990.088 ų, 1228.300 ų and 1125.240 ų for α-quartz, β-quartz, β-cristobalite and α-cristobalite (Table 1), giving an order of bulk volume in β-cristobalite > α-cristobalite > β-quartz > α-quartz, indicating that the bulk volumes in cristobalite phases were greater than that in quartz phases. Moreover, the bulk volume in high temperature SiO₂ phases were higher than that in low temperature SiO₂ phases. In this case, the bulk volume of these typical SiO₂ phases was related to the temperature of the phase transition.

3.2 Evolution of bond length

The substitution of Si⁴⁺ by Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ were performed to investigate the influence of impurity ion to the plastic deformation process from α-quartz to α-cristobalite. The structures of original and substitution 2 × 2 × 2 α-quartz, 2 × 2 × 2 β-quartz, 3 × 1 × 1 β-cristobalite and 3 × 2 × 1 α-cristobalite were shown in Fig. 2, S1, S2 and S3,† respectively. From Fig. 2, four Ti–O, Al–O and H–O bonds were formed when Si⁴⁺ was substituted by Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺, while the Li in the interstitial of the lattice did not bonded with the surrounding O atoms (Fig. 2(c)). In addition, the changes of structure caused by impurity ion substitution (Fig. 2) were significant smaller than that caused by phase transition (Fig. 1), the impurity substitution only affected the length of the surrounding Si–O bonds. Similar structure changes were observed in β-quartz, β-cristobalite and α-cristobalite, thus the bond lengths in these original and substitution typical SiO₂ phases were investigated to clarify the plastic deformation caused by impurity ion substitution and phase transition.

Fig. 2 Structures of optimized original and Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ substituted 2 × 2 × 2 α-quartz supercells: (a) Si₂₄O₄₈; (b) Si₂₃TiO₄₈; (c) Si₂₃AlLiO₄₈; (d) Si₂₃H₄O₄₈.

The average bond lengths of Si–O, Ti–O, Al–O and H–O bonds in original and substituted SiO₂ phases were shown in Fig. 3. Overall, the average bond lengths were in an order of Ti–O > Al–O > Si–O > H–O bonds, since the ionic radius for Ti⁴⁺, Al³⁺, Si⁴⁺ and H⁺ were 0.61 Å, 0.54 Å, 0.40 Å and 0.24 Å.^18,31 For the Si–O bonds, the average Si–O bond lengths for α-quartz, β-quartz, β-cristobalite and α-cristobalite in the original phases were 1.622 Å, 1.614 Å, 1.608 Å and 1.617 Å, respectively. It was clearly that the average Si–O bond lengths in β-quartz (or β-cristobalite) were shorter than that in α-quartz (or α-cristobalite).¹⁶ Same evolution rules for average Si–O bond lengths were presented in Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ substituted SiO₂ phases, indicating that the average Si–O bond lengths in high temperature were shorter than that in low temperature for all original and substituted SiO₂ phases. In addition, the average Si–O bond length changes caused by phase transition were more significant than that due to substitution, e.g., the changes caused by the phase transition were −0.008 Å (i.e., 1.614–1.622 Å), −0.014 Å (i.e., 1.608–1.622 Å) and −0.005 Å (i.e., 1.617–1.622 Å) from α-quartz to β-quartz, β-cristobalite and α-cristobalite, respectively, while caused by Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ substitution for α-quartz were −0.002 Å (i.e., 1.620–1.622 Å), 0.002 Å (i.e., 1.624–1.622 Å) and −0.004 Å (i.e., 1.618–1.622 Å), respectively (Fig. 3(a)).

Fig. 3 Average bond length of original and substituted SiO₂ phases: (a) Si–O bonds; (b) Ti–O bonds; (c) Al–O bonds; (d) H–O bonds.

The evolution rules of Al–O and H–O bonds were similar to Si–O bonds during the plastic deformation process, i.e., the average bond lengths in high temperature SiO₂ phases were shorter than that in low temperature. However, the evolution rule was only applied to the phase transition of α-quartz to β-quartz for the Ti–O bonds. In contrast, the average Ti–O bond length in β-cristobalite was longer than that in α-cristobalite, i.e., the average Ti–O bond lengths were 1.812, 1.800, 1.805 and 1.787 Å for Ti⁴ substitution α-quartz, β-quartz, β-cristobalite and α-cristobalite, respectively, may due to the higher fusion point of Ti than Si and Al (i.e., 1668 °C, 1410 °C and 660 °C for Ti, Si and Al). Overall, the evolution of bond length was mainly dominated by phase transition rather than impurity ion substitution.

3.3 Evolution of volume

Fig. 4 showed the bulk and interstitial volume of original and substituted SiO₂ phases. Both the phase transition and the impurity ion substitution increased the bulk volume. Similar to the original SiO₂ phases as discussed in Section 3.1, the bulk volumes for the phase transformed SiO₂ phases were larger than that of the ion substituted phase, giving an order of bulk volume in β-cristobalite > α-cristobalite > β-quartz > α-quartz, further indicating greater bulk volume in high temperature SiO₂ phases than that in low temperature for all SiO₂ phases. For instance, the bulk volume changes caused by phase transition were 56.864 ų (i.e., 990.088–933.224 ų), 295.076 ų (i.e., 1228.300–933.224 ų) and 192.016 ų (i.e., 1125.240–933.224 ų) from α-quartz to β-quartz, α-cristobalite and β-cristobalite, while the bulk volume changes caused by Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ substitution for α-quartz were 43.336 ų (i.e., 976.560–933.224 ų), 51.130 (i.e., 984.354–933.224 ų) and 55.651 ų (i.e., 988.875–933.224 ų), respectively. In this case, the evolution of bulk volume was mainly dominated by phase transition rather than substitution due to impurity ions.

Fig. 4 (a) Bulk volume and (b) interstitial volume of original and substituted SiO₂ phases during phase transition process.

The interstitial volume was closely related to the migration of interstitial impurity such as Li⁺, Na⁺, K⁺, etc.¹⁶ Thus the evolution of interstitial volume was calculated (Fig. 4(b)). Similar to the evolution of bulk volume, the interstitial volume changes caused by phase transition were more significant than that by impurity ion substitution, e.g., the interstitial volume changes caused by phase transition were 59.049 ų (i.e., 937.000–877.951 ų), 299.133 ų (i.e., 1177.084–877.951 ų) and 194.969 ų (i.e., 1072.920–877.951 ų) from α-quartz to β-quartz, α-cristobalite and β-cristobalite, respectively, while caused by Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ substitution for α-quartz were 47.232 ų (i.e., 925.183–877.951 ų), 53.968 ų (i.e., 931.919–877.951 ų) and 61.267 ų (i.e., 939.218–877.951 ų), respectively (Fig. 3(b)). These indicated that the evolution of interstitial volume was mainly dominated by phase transition rather than substitution due to impurity ions.

In addition, the interstitial volume changes caused by the phase transition was greater than that of bulk volume, possibly due to bigger bulk volume and shorter average bond lengths for the high temperature SiO₂ phases. Moreover, the interstitial volume change from β-quartz to α-cristobalite was significant greater than that from α-quartz to β-quartz, i.e., 240.084 ų (i.e., 1177.084–937.000 ų) vs. 59.049 ų, indicating that the migration of interstitial impurities was proportional to temperature.¹⁶

3.4 Evolution of lattice parameter

Since the evolution of bulk volume was mainly dominated by phase transition, the changes of lattice parameters (Δa_i, Δb_i and Δc_i) were employed to evaluate the contribution of lattice parameters to the changes of bulk volume (ΔV). As shown in Fig. 5(a) the Δa_i increased with the increase of ΔV for original SiO₂ phases, while Δb_i and Δc_i decreased with the increased ΔV, indicating that Δa_i positively contributed to the increase of bulk volume. In contrast, Δb_i and Δc_i exhibited negative contributes. The values of Δa_i/ΔV, Δb_i/ΔV and Δc_i/ΔV of 0.0370, −0.0052 and −0.0138 indicated that the bulk volume of original SiO₂ phases were increased via a axis, but decreased mainly via c axis during the phase transition process.²⁶

Fig. 5 The contribution of Δa_i, Δb_i and Δc_i to ΔV in (a) original, (b) Ti⁴⁺ substituted, (c) Al³⁺/Li⁺ substituted and (d) 4H⁺ substituted SiO₂ phases.

Similar contribution rules were observed for Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ substitution, i.e., the bulk volume of substituted SiO₂ phases increased via a axis, but decreased mainly via c axis during the phase transition process. In this case, the substitution of impurity ion did not change the contribution rule between lattice parameters to bulk volume, but changing the specific Δa_i/ΔV, Δb_i/ΔV and Δc_i/ΔV values. For instance, the Δa_i/ΔV values in Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ substituted SiO₂ phases (i.e., 0.0404, 0.0446 and 0.0458, respectively) were greater than that of original SiO₂ phases (i.e., 0.0370), giving a Δa_i/ΔV value order in 4H⁺ > Al³⁺/Li⁺ > Ti⁴⁺ > original SiO₂ phases. Same orders were observed for the absolute values of Δb_i/ΔV and Δc_i/ΔV, indicating that the contribution of lattice parameters to bulk volumes was related to the number of substitution atoms.

3.5 Evolution of lattice stress

Normally, the change of bulk volume needs to overcome the lattice stress, thus lattice stress of original and substituted SiO₂ phases were calculated based on eqn (1). Overall, the lattice stresses were negative for original and substituted β-quartz and β-cristobalite (Fig. 6), but positive for α-cristobalite, e.g., the lattice stresses were −10.521 GPa, −34.095 GPa and 6.740 GPa for original β-quartz, β-cristobalite and α-cristobalite, indicating that the phase transition from α-quartz to β-quartz and from β-quartz to α-cristobalite occurred spontaneously. However, the evolution from α-cristobalite to β-cristobalite needs to overcome lattice stress.

Fig. 6 Lattice stress of original and substituted SiO₂ phases during phase transition. The ΔV used in the lattice stress calculation for β-quartz, β-cristobalite and α-cristobalite were volume changes from α-quartz to β-quartz, from β-quartz to β-cristobalite and from β-cristobalite to α-cristobalite, respectively.

Furthermore, the substitution of impurity ion did not change the lattice stress of SiO₂ phases during the phase transition process, but changing the specific lattice stress values. For instance, the lattice stresses for original, Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ substituted β-quartz were −10.521 GPa, −19.246 GPa, −8.454 GPa and −12.356 GPa, respectively, indicating that the substitution of Ti⁴⁺ and 4H⁺ drived the phase transition from α-quartz to β-quartz easily. However, the substitution of Al³⁺/Li⁺ occurred difficultly. Differently, the substitution of Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ promoted the phase transition from β-quartz to α-cristobalite easily. In contrast, the phase transition from α-cristobalite to β-cristobalite need to overcome greater lattice stress due to the substitution of Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺.

3.6 Evolution of energy

In order to better reveal the energy evolution during the phase transition process, the energy barriers were calculated (Fig. 7(a) and Tables S1–S5†). It was clearly that the energy barriers for β-quartz and β-cristobalite were positive, but negative for α-cristobalite in all SiO₂ phases, e.g. the energy barriers were 0.489, 0.333 and −1.973 eV for original β-quartz, β-cristobalite and α-cristobalite, respectively, indicating that the phase transitions from α-quartz to β-quartz and from β-quartz to α-cristobalite need to overcome energy barriers. In contrast, the phase transition from α-cristobalite to β-cristobalite occurred spontaneously from thermodynamic point of view. In other words, the phase transition from α-quartz to β-quartz and further to α-cristobalite were endothermic process but the phase transition from α-cristobalite to β-cristobalite was exothermic process, consistent with the phase transition temperature results shown in Fig. 1.

Fig. 7 (a) Energy barrier and (b) formation energy of original and substituted SiO₂ phases during phase transition.

In addition, the substitution of lattice impurity significantly reduced the energy barriers for the phase transition from α-quartz to β-quartz and further to α-cristobalite, e.g., the energy barriers for original, Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ substituted SiO₂ phases were 0.489, 0.210, 0.147 and 0.115 eV, respectively, indicating that the present of lattice impurities promoted phase transition from thermodynamic point of view.

The formation energy of original and substituted SiO₂ phases were calculated to evaluate the stability of lattice impurity substituted structure (Fig. 7(b)). The original, Ti⁴⁺ and Al³⁺/Li⁺ substituted SiO₂ phases exhibited negative formation energies, e.g., −2.798, −4.662 and −2.821 eV for α-quartz phases, respectively. However, 4H⁺ substituted SiO₂ phases exhibited positive formation energies, e.g., 9.106 eV for α-quartz, indicating that the original, Ti⁴⁺ and Al³⁺/Li⁺ substituted SiO₂ phases can exist stably, while the 4H⁺ substituted SiO₂ phases were structurally unstable. In addition, the formation energies were in an order of Ti⁴⁺ substitution < Al³⁺/Li⁺ substitution < original < 4H⁺ substitution. In this case, the lattice impurities Ti and Al were hardly to be removed due to their lower formation energies than that of original SiO₂ phases.¹⁸

4 Conclusions

This study investigated the plastic deformation mechanisms for the phase transition between α-quartz, β-quartz, β-cristobalite and α-cristobalite based on the evolution of supercell structures, bond lengths, volumes, lattice parameters, lattice stresses and energies in original, Ti⁴⁺, Al³⁺/Li⁺ and 4H⁺ substituted SiO₂ phases. Overall, the evolution of bond lengths and volumes were mainly dominated by phase transition rather than substitution due to impurity ions. The average Si–O bond lengths in high temperature SiO₂ phases were shorter than that in low temperature for original and substituted SiO₂ phases. In contrast, the bulk volume in high temperature SiO₂ phases were higher than that in low temperature. Thus, the interstitial volume changes caused by the phase transition was greater than that of the bulk volume, beneficial to the migration of interstitial impurities. In addition, the bulk volume of original and substituted SiO₂ phases increased via a axis, but decreased mainly via c axis during the phase transition process. In addition, the contribution of lattice parameters to bulk volumes was related to the number of substitution atoms. Moreover, the phase transition from α-quartz to β-quartz and further to α-cristobalite need to overcome energy barriers, while the evolution from α-cristobalite to β-cristobalite needs to overcome a lattice stress.

Author contributions

Methodology, Zhenlun wei, Wanqing Li, Xianglin Hu, software, Zhenlun Wei, Xianglin Hu, Yunxiang Gu, writing—original draft preparation, Zhenlun Wei, Peiyue Li, Li Pan, writing—review and editing, Yubiao Li, supervision, Yubiao Li.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

The authors would like to acknowledge the financial support from the Key Research and Development Program of Hubei Province (2021BCA127).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3ra07633d

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