Constraints on the phase diagram of molybdenum from first-principles free-energy calculations

Physical Review B - Condensed Matter and Materials Physics

Volumn 85, Issue 6, 2012, Pages

  Cazorla, C ^a,b   Alfe D ^b   Gillan, M J ^b  

^a UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

^b UNIVERSITY COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84857697866     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.85.064113     Document Type: Article

References (70)

1. C. Cazorla, M. J. Gillan, S. Taioli, and D. Alfè, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.2735324 126, 194502 (2007). (Pubitemid 46794135)
2. S. Taioli, C. Cazorla, M. J. Gillan, and D. Alfè, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.75.214103 75, 214103 (2007).
3. D. Errandonea, B. Schwager, R. Ditz, C. Gessman, R. Boehler, and M. Ross, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.63.132104 63, 132104 (2001).
4. D. Alfè, G. D. Price, and M. J. Gillan, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.65.165118 65, 165118 (2002).
5. D. Errandonea, Physica B PHYADX 0378-4371 10.1016/j.physa.2005.06.036 357, 356 (2005). (Pubitemid 41327893)
6. D. Santamaria-Perez, M. Ross, D. Errandonea, G. D. Mukherjee, M. Mezouar, and R. Boehler, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.3082030 130, 124509 (2009).
7. A. B. Belonoshko, L. Burakovsky, S. P. Chen, B. Johansson, A. S. Mikhaylushkin, D. L. Preston, S. I. Simak, and D. C. Swift, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.100.135701 100, 135701 (2008). (Pubitemid 351490841)
8. A. S. Mikhaylushkin, S. I. Simak, L. Burakovsky, S. P. Chen, B. Johansson, D. L. Preston, D. C. Swift, and A. B. Belonoshko, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.101.049602 101, 049602 (2008).
9. T. S. Duffy, Rep. Prog. Phys. RPPHAG 0034-4885 10.1088/0034-4885/68/8/R03 68, 1811 (2005). (Pubitemid 41024592)
10. L. Burakovsky, S. P. Chen, D. L. Preston, A. B. Belonoshko, A. Rosengren, A. S. Mikhaylushkin, S. I. Simak, and J. A. Moriarty, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.104.255702 104, 255702 (2010).
11. A. Dewaele, M. Mezouar, N. Guignot, and P. Loubeyre, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.104.255701 104, 255701 (2010).
12. C. J. Wu, P. Söderlind, J. N. Glosli, and J. E. Klepeis, Nat. Mater. 1476-1122 10.1038/nmat2375 8, 223 (2009).
13. Z.-L Liu, L.-C. Cai, X.-R. Chen, and F.-Q. Jing, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.77.024103 77, 024103 (2008).
14. R. S. Hixson and J. N. Fritz, J. Appl. Phys. JAPIAU 0021-8979 10.1063/1.351203 71, 1721 (1992).
15. A. C. Mitchell and W. J. Nellis, J. Appl. Phys. JAPIAU 0021-8979 10.1063/1.329160 52, 3363 (1981). (Pubitemid 12430229)
16. C. Cazorla, D. Alfè, and M. J. Gillan, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.101.049601 101, 049601 (2008).
17. Z.-Y. Zeng, C.-E. Hu, L.-C. Cai, X.-R. Chen, and F.-Q. Jing, J. Phys. Chem. B JPCBFK 1520-6106 10.1021/jp9073637 114, 298 (2010).
18. S. Japel, B. Schwager, R. Boehler, and M. Ross, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.95.167801 95, 167801 (2005). (Pubitemid 41777469)
19. D. Errandonea, M. Somayazulu, D. Hausermann, and H. K. Mao, J. Phys. Condens. Matter JCOMEL 0953-8984 10.1088/0953-8984/15/45/003 15, 7635 (2003).
20. A. B. Belonoshko, S. I. Simak, A. E. Kochetov, B. Johansson, L. Burakovsky, and D. L. Preston, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.195701 92, 195701 (2004).
21. L. Stixrude, R. E. Cohen, and D. J. Singh, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.50.6442 50, 6442 (1994).
22. L. Vočadlo, J. Brodholt, D. Alfè, M. J. Gillan, and G. D. Price, Phys. Earth Planet. Inter. PEPIAM 0031-9201 10.1016/S0031-9201(99)00092-8 117, 123 (2000). (Pubitemid 30105176)
23. S. Taioli, C. Cazorla, M. J. Gillan, and D. Alfè, J. Phys.: Conf. Series 1742-6588 10.1088/1742-6596/121/1/012010 121, 012010 (2008).
24. C. Cazorla, M. J. Gillan, S. Taioli, and D. Alfè, J. Phys.: Conf. Series 1742-6588 10.1088/1742-6596/121/1/012009 121, 012009 (2008).
25. D. Alfè, G. Kresse, and M. J. Gillan, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.61.132 61, 132 (2000).
26. L. Vočadlo, I. G. Wood, M. J. Gillan, J. Brodholt, D. P. Dobson, G.D. Price, and D. Alfè, Phys. Earth Planet. Inter. PEPIAM 0031-9201 10.1016/j.pepi.2008.07.032 170, 52 (2008).
27. H. K. Mao, J. Xu, V. V. Struzhkin, J. Shu, R. J. Hemley, W. Sturhahn, M. Y. Hu, E. E. Alp, L. Vocadlo, D. Alfé, G. D. Price, M. J. Gillan, M. Schwoerer-Böhning, D. Häusermann, P. Eng, G. Shen, H. Giefers, R. Lübbers, and G. Wortmann, Science SCIEAS 0036-8075 10.1126/science.1057670 292, 914 (2001). (Pubitemid 32422485)
28. C. Cazorla, D. Alfè, and M. J. Gillan, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.77.224103 77, 224103 (2008).
29. G. A. de Wijs, G. Kresse, and M. J. Gillan, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.57.8223 57, 8223 (1998).
30. D. Alfè, M. J. Gillan, and G. D. Price, Nature (London) NATUAS 0028-0836 10.1038/46758 401, 462 (1999). (Pubitemid 29470200)
31. A. Dewaele, M. Mezouar, N. Guignot, and P. Loubeyre, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.76.144106 76, 144106 (2007).
32. L. Vočadlo and D. Alfè, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.65.214105 65, 214105 (2002).
33. O. Sugino and R. Car, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.74.1823 74, 1823 (1995).
34. M. J. Gillan, D. Alfè, J. Brodholt, L. Vočadlo, and G. D. Price, Rep. Prog. Phys. RPPHAG 0034-4885 10.1088/0034-4885/69/8/R03 69, 2365 (2006). (Pubitemid 44144176)
35. L. Vočadlo, D. Alfè, M. J. Gillan, and G. D. Price, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.1640344 120, 2872 (2004).
36. S. M. Davis, A. B. Belonoshko, B. Johansson, N. V. Skorodumova, and A. C. T. van Duin, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.3013704 129, 194508 (2008).
37. A. B. Belonoshko, N. V. Skorodumova, A. Rosengren, and B. Johansson, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.73.012201 73, 012201 (2006). (Pubitemid 43270414)
38. O. H. Nielsen and R. M. Martin, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.32.3792 32, 3792 (1985).
39. D. Alfè and M. J. Gillan, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.81.5161 81, 5161 (1998).
40. L. Vočadlo, D. Alfè, M. J. Gillan, I. G. Wood, J. P. Brodholt, and G. D. Price, Nature (London) NATUAS 0028-0836 10.1038/nature01829 424, 536 (2003). (Pubitemid 36975772)
41. D. Alfè, L. Vočadlo, M. J. Gillan, and G. D. Price, J. Phys. Condens. Matter JCOMEL 0953-8984 10.1088/0953-8984/16/14/006 16, S973 (2004).
42. D. Alfè, M. J. Gillan, and G. D. Price, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.1460865 116, 6170 (2002). (Pubitemid 34476575)
43. P. E. Blöchl, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.50. 17953 50, 17953 (1994).
44. G. Kresse and J. Furthmüller, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.54.11169 54, 11169 (1996).
45. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.77.3865 77, 3865 (1996).
46. H. J. Monkhorst and J. D. Pack, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.13.5188 13, 5188 (1976).
47. N. D. Mermin, Phys. Rev. A PLRAAN 0031-899X 10.1103/PhysRev.137.A1441 137, 1441 (1965).
48. D. Alfè, G. D. Price, and M. J. Gillan, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.64.045123 64, 045123 (2001).
49. A. R. Oganov and P. I. Dorogokupets, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.67.224110 67, 224110 (2003).
50. A. R. Oganov and P. I. Dorogokupets, J. Phys. Condens. Matter JCOMEL 0953-8984 10.1088/0953-8984/16/8/018 16, 1351 (2004).
51. A. B. Belonoshko, P. I. Dorogokupets, B. Johansson, S. K. Saxena, and L. Koci, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.78.104107 78, 104107 (2008).
52. G. Kresse, J. Furthmüller, and J. Hafner, Europhys. Lett. EULEEJ 0295-5075 10.1209/0295-5075/32/9/005 32, 729 (1995).
53. D. Alfè, program is available at [http://chianti.geol.ucl.ac.uk/ ~dario] (1998);
54. D. Alfè, Comput. Phys. Commun. CPHCBZ 0010-4655 10.1016/j.cpc.2009.03.010 180, 2622 (2009).
55. F. Birch, J. Geophys. Res. JGREA2 0148-0227 10.1029/JB083iB03p01257 83, 1257 (1978).
56. C. Asker, A. B. Belonoshko, A. S. Mikhaylushkin, and I. A. Abrikosov, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.77.220102 77, 220102 (R) (2008).
57. We note here the caveat that should be borne in mind for all types of thermodynamic integration; caution should be exercised if the path of integration passes through regions of thermodynamically unstable states or if the thermally averaged integrand (stress, in the present case) is a discontinuous function of the integration variable (strain). The results we present show that the thermally averaged stress tensor varies continuously along the Bain path, so that there is no indication of instabilities on the path. Furthermore, the correctness of Bain-path integration will be confirmed a posteriori by the fact that the free-energy differences deduced from it are corroborated by explicit calculation of the anharmonic free energies (see Sec.).
58. M. S. Daw and M. I. Baskes, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.29.6443 29, 6443 (1984).
59. M. W. Finnis and J. E. Sinclair, Philos. Mag. A PMAADG 0141-8610 10.1080/01418618408244210 50, 45 (1984).
60. We have also checked the effect on the fcc-bcc free energy difference when we reduce the PAW core radius. We reduced the core radius from 2.7 a.u. to 1.6 a.u. and we used first-order perturbation theory to estimate the changes of free energies of fcc and bcc at the state V = 8.26 3 /atom and T = 9000 K. We found that these free energy changes amount to only ∼ 10 meV/atom, which is much smaller than our computed free energy differences between fcc and bcc.
61. We note that the use of harmonic reference systems in calculating the free energy of anharmonic systems is very well established. A few of the many examples are D. Frenkel and A. J. C. Ladd, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.448024 81, 3188 (1984);
62. S. M. Foiles, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.49.14930 49, 14930 (1994);
63. O. Sugino and R. Car, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.74.1823 74, 1823 (1995);
64. U. Hansen, P. Vogl, and V. Fiorentini, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.60.5055 60, 5055 (1999).
65. D. Alfè, Phys. Rev. B PLRBAQ 1098-0121 10.1103/PhysRevB.79.060101 79, 060101 (R) (2009).
66. E. Sola and D. Alfè, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.103.078501 103, 078501 (2009).
67. D. Alfè, C. Cazorla, and M. J. Gillan, J. Chem. Phys. 0021-9606 10.1063/1.3605601 135, 024102 (2011).
68. V. Ozolins, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.102. 065702 102, 065702 (2009).
69. This point was emphasised by the originators of the Z method, noting in Ref. 37 that "... in this method both a rather large number of atoms and long runs are needed... ".
70. C. Cazorla, D. Alfè, and M. J. Gillan, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.3126683 130, 174707 (2009).