Skip to main content

Advertisement

SpringerLink
Log in

A comprehensive study of the high-pressure–temperature phase diagram of silicon

  • Computation
  • Published:
Journal of Materials Science Aims and scope Submit manuscript

Abstract

A high-pressure–temperature (P–T) phase diagram of silicon (Si) has been constructed in a range of temperatures covering from 0 to 2000 K and pressures up to 80 GPa. In this system, there is a potential for the following phases to occur: cubic diamond, β-tin, simple hexagonal, double-hexagonal close-packed, hexagonal close-packed and face-centered cubic. Besides, the lattice vibrational energy and thermodynamic quantities of each phase were calculated, in combination with quasi-harmonic approximation using the first-principles phonon density of state. The calculated temperature dependencies of thermodynamic quantities are in good agreement with experimental and theoretical observations. Thereinto, the Gibbs free energy differences of these six phases, as a function of pressure and temperature, were used to define the phase boundaries of the P–T phase diagram of Si. The results in this work not only integrate previous experimental and theoretical investigations, but also successfully predict the phase relationships of the stable phases of Si under conditions of high temperature and pressure for the first time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9

Similar content being viewed by others

References

  1. Kodiyalam S, Kalia RK, Kikuchi H, Nakano A, Shimojo F, Vashishta P (2001) Grain boundaries in gallium arsenide nanocrystals under pressure: a parallel molecular-dynamics study. Phys Rev Lett 86:55–58

    Article  Google Scholar 

  2. Malone BD, Louie SG, Cohen ML (2010) Electronic and optical properties of body-centered-tetragonal Si and Ge. Phys Rev B 81:115201

    Article  Google Scholar 

  3. Georg RB, Halliday AN, Schauble EA, Reynolds BC (2007) Silicon in the earth’s core. Nature 447:1102–1106

    Article  Google Scholar 

  4. Hanflan M, Schwarz U, Syassen K, Takemura K (1999) Crystal structure of the high-pressure phase silicon VI. Phys Rev Lett 82:1197–1200

    Article  Google Scholar 

  5. Minomura S, Drickamer HG (1962) Pressure induced phase transitions in Silicon, Germanium and some III–V compounds. J Phys Chem Solids 23:451–456

    Article  Google Scholar 

  6. Jamieson JC (1963) Crystal structures at high pressures of metallic modifications of silicon and germanium. Science 139:762–764

    Article  Google Scholar 

  7. Bundy FP (1964) Phase diagrams of Silicon and Germanium to 200 kbar, 1000  C. J Chem Phys 41:3809–3814

    Article  Google Scholar 

  8. Welber B, Kim CK, Cardona M, Rodriguez S (1975) Dependence of the indirect energy gap of silicon on hydrostatic pressure. Solid State Commun 17:1021–1024

    Article  Google Scholar 

  9. Hu JZ, Merkle LD, Menoni CS, Spain IL (1986) Crystal data for high-pressure phases of silicon. Phys Rev B 34:4679–4684

    Article  Google Scholar 

  10. Olijnyk H, Sikka SK, Holzapfel WB (1984) Structural phase transitions in Si and Ge under pressures up to 50 GPa. Phys Lett A 103:137–140

    Article  Google Scholar 

  11. Hu JZ, Spain IL (1984) Phases of silicon at high pressure. Solid State Commun 51:263–266

    Article  Google Scholar 

  12. Jayaraman A, Klement W, Kennedy GC (1963) Melting and polymorphism at high pressures in some group IV elements and III–V compounds with the diamond/zincblende structure. Phys Rev 130:540–547

    Article  Google Scholar 

  13. Duclos SJ, Vohra YK, Ruoff AL (1990) Experimental study of the crystal stability and equation of state of Si to 248 GPa. Phys Rev B 41:12021–12028

    Article  Google Scholar 

  14. Duclos SJ, Vohra YK, Ruoff AL (1987) Hcp to fcc transition in Silicon at 78 GPa and studies to 100 GPa. Phys Rev Lett 58:775–777

    Article  Google Scholar 

  15. Ahuja R, Eriksson O, Johansson B (1999) Theoretical high-pressure studies of silicon VI. Phys Rev B Condens Matter Mater Phys 60:475–477

    Article  Google Scholar 

  16. Needs RJ, Mujica A (1995) A first-principles pseudopotential dtudy of the structural phases of Silicon. Phys Rev B 51:9652–9660

    Article  Google Scholar 

  17. Kasper JS, Richards SM (1964) The crystal structures of new forms of silicon and germanium. Acta Cryst 17:752–755

    Article  Google Scholar 

  18. Besson JM, Mokhtari EH, Gonzalez J, Weill G (1987) Electrical properties of demimetallic silicon III and semiconductive silicon IV at ambient pressure. Phys Rev Lett 59:473–476

    Article  Google Scholar 

  19. Zhao YX, Buehler F, Sites JR, Spain IL (1986) New metastable phases of Silicon. Solid State Commun 59:679–682

    Article  Google Scholar 

  20. Piltz RO, Maclean JR, Clark SJ, Ackland GJ, Hatton PD, Crain J (1995) Structure and properties of silicon XII: a complex tetrahedrally bonded phase. Phys Rev B 52:4072–4085

    Article  Google Scholar 

  21. Crain J, Ackland GJ, Maclean JR, Piltz RO, Hatton PD, Pawley GS (1994) Reversible pressure-induced structural transitions between metastable phases of silicon. Phys Rev B 50:13043–13046

    Article  Google Scholar 

  22. Pfrommer BG, Côté M, Louie SG, Cohen ML (1997) Ab initio study of Silicon in the R8 phase. Phys Rev B 56:6662–6668

    Article  Google Scholar 

  23. Nguyen MC, Zhao X, Wang Y, Wang CZ, Ho KM (2014) Genetic algorithm prediction of crystal structure of metastable Si-IX phase. Solid State Commun 182:14–16

    Article  Google Scholar 

  24. Goettel KA, Mao HK, Bell PM (1985) Generation of static pressures above 2.5 megabars in a diamond-anvil pressure cell. Rev Sci Instrum 56:1420–1427

    Article  Google Scholar 

  25. Domnich V, Gogotsi Y (2002) Phase transformations in Silicon under contact loading. Rev Adv Mater Sci 3:1–36

    Article  Google Scholar 

  26. Broughton JQ, Li XP (1987) Phase diagram of silicon by molecular dynamics. Phys Rev B 35:9120–9127

    Article  Google Scholar 

  27. Hebbache M, Mattesini M, Szeftel J (2001) Pressure-induced structural sequence in silicon: diamond to β-Tin to imma. Phys Rev B 63:205201

    Article  Google Scholar 

  28. Yang CC, Li JC, Jiang Q (2003) Effect of pressure on melting temperature of silicon determined by Clapeyron equation. Chem Phys Lett 372:156–159

    Article  Google Scholar 

  29. Gaál-Nagy K, Bauer A, Pavone P, Strauch D (2004) Ab initio study of the enthalpy barriers of the high-pressure phase transition from the cubic-diamond to the β-Tin structure of Silicon and Germanium. Comput Mater Sci 30:1–7

    Article  Google Scholar 

  30. Yang CC, Li JC, Jiang Q (2004) Temperature-pressure phase diagram of silicon determined by Clapeyron equation. Solid State Commun 129:437–441

    Article  Google Scholar 

  31. Kaczmarski M, Bedoya-Martinez ON, Hernandez ER (2005) Phase diagram of silicon from atomistic simulations. Phys Rev Lett 94:095701

    Article  Google Scholar 

  32. Wang CP, Li C, Han JJ, Yan LH, Deng B, Liu XJ (2017) The pressure-temperature phase diagram of pure Co based on first-principles calculations. Phys Chem Chem Phys 19:22061–22068

    Article  Google Scholar 

  33. Mei ZG, Shang SL, Wang Y, Liu ZK (2009) Density-functional study of the thermodynamic properties and the pressure-temperature phase diagram of Ti. Phys Rev B 80:104116

    Article  Google Scholar 

  34. Lee C, Gonze X (1995) Ab initio calculation of the thermodynamic properties and atomic temperature factors of SiO2 α-quartz and stishovite. Phys Rev B 51:8610–8613

    Article  Google Scholar 

  35. Mermin ND (1965) Thermal properties of the inhomogeneous electron gas. Phys Rev 137:A1441–A1443

    Article  Google Scholar 

  36. Kresse G (1998) From ultrasoft pseudopotentials to the projector augmented-wave method. Phys Rev B 59:1758–1775

    Article  Google Scholar 

  37. Kresse G, Furthmüller J (1996) Efficiency of ab initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput Mater Sci 6:15–50

    Article  Google Scholar 

  38. Perdew J, Burke K, Ernzerhof M (1996) K + emission in symmetric heavy ion reactions at subthreshold energies. Phys Rev Lett 78:1396

    Article  Google Scholar 

  39. Blöchl PE (1994) Projector augmented-wave method. Phys Rev B 50:17953–17979

    Article  Google Scholar 

  40. Monkhorst HJ, Pack JD (1976) Special points for Brillouin-zone integrations. Phys Rev B 13:5188–5192

    Article  Google Scholar 

  41. Baroni S, Gironcoli SD, Corso AD, Giannozzi P (2001) Phonons and related crystal properties from density-functional. Rev Mod Phys 73:515–562

    Article  Google Scholar 

  42. Togo A, Oba F, Tanaka I (2008) First-principles calculations of the ferroelastic transition between rutile-type and CaCl2-type SiO2 at high pressures. Phys Rev B 78:134106

    Article  Google Scholar 

  43. Togo A, Chaput L, Tanaka I, Hug G (2010) First-principles phonon calculations of thermal expansion in Ti3SiC2, Ti3AlC2, and Ti3GeC2. Phys Rev B 81:174301

    Article  Google Scholar 

  44. Liu ZL (2015) Phasego: a toolkit for automatic calculation and plot of phase diagram. Comput Phys Commun 191:150–158

    Article  Google Scholar 

  45. Liu ZL (2015) Phasego 2.0: counting full anharmonic effects from high-temperature phonon density of states. Comput Phys Commun 197:341–342

    Article  Google Scholar 

  46. Liu ZL, Wang HY, Li XF (2016) Phasego 3.0: automatic analysis of synthesis and decomposition. Comput Phys Commun 29:197–198

    Article  Google Scholar 

  47. Birch F (1978) Finite strain isotherm and velocities for single-crystal and polycrystalline NaCl at high pressures and 300 K. J Geophys Res Solid Earth 83:1257–1268

    Article  Google Scholar 

  48. Baonza VG, Cáceres M, Núñez J (1995) Universal compressibility behavior of dense phases. Phys Rev B 51:28–37

    Article  Google Scholar 

  49. Cannon JF (1974) Behavior of the elements at high pressures. J Phys Chem Ref Data 3:781–824

    Article  Google Scholar 

  50. Yin MT, Cohen ML (1984) Structural theory of graphite and graphitic silicon. Phys Rev B 29:6996–6998

    Article  Google Scholar 

  51. Chang KJ, Cohen ML (1985) Solid-solid phase transitions and soft phonon modes in highly condensed Si. Phys Rev B 31:7819–7826

    Article  Google Scholar 

  52. Needs RJ, Martin RM (1984) Transition from β-tin to simple hexagonal Silicon under pressure. Phys Rev B 30:5390–5392

    Article  Google Scholar 

  53. Malone BD, Cohen ML (2012) Prediction of a metastable phase of Silicon in the Ibam structure. Phys Rev B 85:024116

    Article  Google Scholar 

  54. Nilsson G, Nelin G (1972) Study of the homology between silicon and germanium by thermal-neutron spectrometry. Phys Rev B 6:3777–3786

    Article  Google Scholar 

  55. Dolling G, Cowley RA (1966) The thermodynamic and optical properties of germanium, silicon, diamond and gallium. Proc Phys Soc 88:463–494

    Article  Google Scholar 

  56. Wakabayashi N, Scherm RH, Smith HG (1982) Lattice dynamic of Ti Co, Tc, and other hcp transition metals. Phys Rev B 25:5122–5132

    Article  Google Scholar 

  57. Ekman M, Persson K, Grimvall G (2000) Lattice dynamics and thermodynamic properties of the β-Sn phase in Si. Phys Rev B 62:14784–14789

    Article  Google Scholar 

  58. Tse JS, Klug DD, Patchkovskii S, Ma Y, Dewhurst JK (2006) Chemical bonding, electron-phonon coupling, and structural transformations in high-pressure phases of Si. J Phys Chem B 110:3721–3726

    Article  Google Scholar 

  59. Giannozzi P, de Gironcoli S, Pavone P, Baroni S (1991) Ab initio calculation of phonon dispersions in semiconductors. Phys Rev B 43:7231–7242

    Article  Google Scholar 

  60. Wei S, Chou MY (1994) Phonon dispersions of silicon and germanium from first-principles calculations. Phys Rev B 50:2221–2226

    Article  Google Scholar 

  61. Favot F, Corso AD (1999) Phonon dispersions performance of the generalized gradient approximation. Phys Rev B 60:11427

    Article  Google Scholar 

  62. Carr RH, McCammon RD, White GK (1965) Thermal expansion of germanium and silicon at low temperatures. Philos Mag 12:157–163

    Article  Google Scholar 

  63. Ibach H (1969) Thermal expansion of Silicon and Zinc Oxide. Phys Stat Solid 31:625–634

    Article  Google Scholar 

  64. Yim WM, Paff RJ (1974) Thermal expansion of AlN, sapphire, and silicon. J Appl Phys 45:1456–1457

    Article  Google Scholar 

  65. Roberts RB (1981) Thermal expansion reference data Silicon 300–850 K. J Phys D Appl Phys 14:163–166

    Article  Google Scholar 

  66. Lyon KG, Salinger GL, Swenson CA, White GK (1977) Linear thermal expansion measurements on Silicon from 6 to 340 K. J Appl Phys 48:865–868

    Article  Google Scholar 

  67. Xu CH, Wang CZ, Chan CT, Ho KM (1991) Theory of the thermal expansion of Si and diamond. Phys Rev B 43:5024–5027

    Article  Google Scholar 

  68. Yin MT, Cohen ML (1980) Theory of the phase transformation and lattice dynamics of Si. Phys Rev Lett 45:1004–1007

    Article  Google Scholar 

  69. Flubacher P, Leadbetter AJ, Morrison JA (1959) The heat capacity of pure silicon and germanium and properties of their vibrational frequency spectra. Philos Mag 4:273–294

    Article  Google Scholar 

  70. Chase MW, Davies CA, Downey JR, Frurip DJ, Mcdonald RA, Syverud AN (1985) Janaf thermochemical table. J Phys Chem Ref Data 14:1–926

    Article  Google Scholar 

  71. Sharma SM, Sikka SK (1985) Lattice dynamical analysis of β-γ phase transformation in Silicon under high pressure. J Phys Chem Solids 46:477–479

    Article  Google Scholar 

  72. Voronin GA, Pantea C, Zerda TW, Wang L, Zhao Y (2003) In situx-ray diffraction study of Silicon at pressures up to 15.5 GPa and temperatures up to 1073 K. Phys Rev B 68:020102

    Article  Google Scholar 

  73. Kubo A, Wang Y, Runge CE, Uchida T, Kiefer B, Nishiyama N, Duffy TS (2008) Melting curve of silicon to 15 GPa determined by two-dimensional angle-dispersive diffraction using a kawai-type apparatus with x-ray transparent sintered diamond anvils. J Phys Chem Solids 69:2255–2260

    Article  Google Scholar 

  74. Lees J, Williamson BHJ (1965) Combined very high pressure high temperature calibration of the tetrahedral anvil apparatus, fusion curves of zinc, aluminium, germanium and silicon to 60 kilobars. Nature 208:278–279

    Article  Google Scholar 

  75. Brazhkin VV, Lyapin AG, Popova SV, Voloshin RN (1995) Nonequilibrium phase transitions and amorphization in Si, Si/Gaas, Ge, and Ge/Gasb at the decompression of high-pressure phases. Phys Rev B 51:7549–7554

    Article  Google Scholar 

  76. Vechten JAV (1973) Quantum dielectric theory of electronegativity in covalent systems. III. Pressure-temperature phase diagrams, heats of mixing, and distribution coefficients. Phys Rev B 7:1479–1507

    Article  Google Scholar 

  77. Okada Y, Tokumaru Y (1984) Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K. J Appl Phys 56:314–320

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51571168 and 51601160), the National Key R&D Program of China (Grant No. 2017YFB0702901), the Ministry of Science and Technology of China (Grant No. 2014DFA53040) and the Natural Science Foundation of Fujian Province, China (No. 2016J05133).

Author information

Authors and Affiliations

  1. College of Materials and Fujian Provincial Key Laboratory of Materials Genome, Xiamen University, Xiamen, 361005, People’s Republic of China

    Cong Li, Cuiping Wang, Jiajia Han, Lihui Yan, Bin Deng & Xingjun Liu

Authors
  1. Cong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cuiping Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiajia Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lihui Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Bin Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Xingjun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Jiajia Han or Xingjun Liu.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, C., Wang, C., Han, J. et al. A comprehensive study of the high-pressure–temperature phase diagram of silicon. J Mater Sci 53, 7475–7485 (2018). https://doi.org/10.1007/s10853-018-2087-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10853-018-2087-9

Keywords

Search

Navigation