Lambda transitions in materials science: Recent advances in CALPHAD and first-principles modelling

Physica Status Solidi (B) Basic Research

Volumn 251, Issue 1, 2014, Pages 53-80

  Körmann, Fritz ^a   Breidi, Abed Al Hasan ^b   Dudarev, Sergei L ^c   Dupin, Nathalie ^d   Ghosh, Gautam ^e   Hickel, Tilmann ^a   Korzhavyi, Pavel ^f   Muñoz, Jorge A ^g   Ohnuma, Ikuo ^h  

^a MAX PLANCK INSTITUT FÜR EISENFORSCHUNG GMBH   (Germany)

^b RUHR UNIVERSITY BOCHUM   (Germany)

^c CULHAM SCIENCE CENTRE   (United Kingdom)

^d Calcul Thermodynamique   (France)

^e Northwestern University   (United States)

^f ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

^g CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^h TOHOKU UNIVERSITY   (Japan)

Author keywords

CALPHAD; First principles; Lambda transitions; Magnetism; Thermodynamic modelling

Indexed keywords

COMPUTATION THEORY; DEGREES OF FREEDOM (MECHANICS); DENSITY FUNCTIONAL THEORY; MAGNETISM;

CALPHAD; CLUSTER VARIATION METHODS; COMPUTATIONAL TECHNIQUE; FIRST PRINCIPLES; FIRST PRINCIPLES METHOD; LAMBDA TRANSITION; MAGNETIC CONTRIBUTION; THERMODYNAMIC MODELLING;

ORDER DISORDER TRANSITIONS;

EID: 84892455844     PISSN: 03701972     EISSN: 15213951     Source Type: Journal    
DOI: 10.1002/pssb.201350136     Document Type: Article

References (192)

1. W. H. Keesom and A. P. Keesom, Verh. K. Akad. Wet. (Amsterdam) 35, 736 (1932).
2. Q. Chen and B. Sundman, J. Phase Equilib. 22(6), 631-644 (2001).
3. D. de Fontaine, S. G. Fries, G. Inden, P. Miodownik, R. SchmidFetzer, and S. Chen, Calphad 19(4), 499-536 (1995).
4. G. Inden, Proc. Calphad V, 1-13 (1976).
5. R. J. Weiss and K. J. Tauer, Phys. Rev. 102, 1490-1495 (1956).
6. M. Hillert and M. Jarl, Calphad 2(3), 227-238 (1978).
7. G. Inden, Physica 103B, 82 (1981).
8. S. Hertzman and B. Sundman, Calphad 6, 67 (1982).
9. W. Xiong, Q. Chen, P. A. Korzhavyi, and M. Selleby, Calphad 39, 11 (2012).
10. M. van Schilfgaarde, I. A. Abrikosov, and B. Johansson, Nature 400, 46 (1999).
11. M. Palumbo et al., Phys. Status Solidi B, DOI 10.1002/ pssb.201350133 (2013).
12. Y. Tsunoda, J. Phys.: Condens. Matter 3, 7231 (1991).
13. Y. Tsunoda, J. Phys.: Condens. Matter 1(51), 10427 (1989).
14. M. Straub, R. Vollmer, and J. Kirschner, Phys. Rev. Lett. 77, 743 (1996).
15. A. Knappwost and G. Bockstiegel, Z. Elektrochem. 57, 700(1953).
16. S. C. Abrahams, L. Guttman, and J. S. Kasper, Phys. Rev. 127, 2052-2055 (1962).
17. M. Marsman and J. Hafner, Phys. Rev. B 66(22), 224409 (2002).
18. V. Sedov, Sov. Phys. -JETP 15, 88 (1962).
19. H. Umebayashi and Y. Ishikawa, J. Phys. Soc. Jpn. 21, 1281 (1966).
20. Y. Ishikawa and Y. Endoh, J. Phys. Soc. Jpn. 23, 205 (1967).
21. T. Hashimoto and Y. Ishikawa, J. Phys. Soc. Jpn. 23, 213 (1967).
22. Y. Ishikawa and Y. Endoh, J. Appl. Phys. 39(2), 1318-1319 (1968).
23. Y. Endoh and Y. Ishikawa, J. Phys. Soc. Jpn. 30, 205 (1971).
24. R. J. Weiss and K. J. Tauer, Phys. Rev. 102, 1490-1495 (1956).
25. A. Miodownik, Scr. Metall. 3, 931 (1969).
26. Y. Nakamura, M. Shiga, and N. Shikazono, J. Phys. Soc. Jpn. 19, 1177 (1964).
27. Y. Nakamura, Y. Takeda, and M. Shiga, J. Phys. Soc. Jpn. 25, 287 (1968).
28. N. Kamiya, I. Ohnuma, R. Kainuma, and K. Ishida, unpublished.
29. R. Wayne and L. Bartel, Phys. Lett. 28A, 196 (1968).
30. Y. Nakamura, M. Hayase, M. Shiga, Y. Yamamoto, and N. Kawai, J. Phys. Soc. Jpn. 30, 720 (1971).
31. J. Noakes, H. Sato, and A. Arrort, J. Appl. Phys. 42, 1608 (1971).
32. T. Hammerschmidt et al., Phys. Status Solidi B, DOI 10.1002/pssb.201350156 (2013).
33. P. W. Anderson, Phys. Rev. 124, 41-53 (1961).
34. J. Hubbard, Proc. R. Soc. Lond. A, Math. Phys. Sci. 276(1365), 238-257 (1963).
35. G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865-951 (2006).
36. J. Griffith, The Theory of Transition Metal Ions, first ed. (Cambridge University Press, Cambridge, UK, 1961).
37. Z. Rudzikas, Theoretical Atomic Spectroscopy (Cambridge University Press, Cambridge, UK, 1997).
38. P. Fazekas, Electron Correlations and Magnetism (World Scientific, Singapore, 1999.
39. A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B 52, R5467-R5470 (1995.
40. A. Kotani and T. Yamazaki, Prog. Theor. Phys., Suppl. 108, 117 (1992.
41. S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505-1509 (1998).
42. S. L. Dudarev, G. A. Botton, S. Y. Savrasov, Z. Szotek, W. M. Temmerman, and A. P. Sutton Phys. Status Solidi A 166(1), 429-443 (1998).
43. D. Nguyen-Manh and S. L. Dudarev, Phys. Rev. B 80, 104440 (2009).
44. D. J. Kim, New Perspectives in Magnetism of Metals (Springer, New York, 1999).
45. M. Coury, S. Dudarev, W. Foulkes, A. Horsfield, and P. W. Ma, unpublished (2013).
46. P. Fulde, Electron Correlations in Molecules and Solids, third ed., Springer Series in Solid-State Sciences, Vol. 100 (Springer, Berlin, 1995).
47. A. Liechtenstein, M. Katsnelson, V. Antropov, and V. Gubanov, J. Magn. Magn. Mater. 67(1), 65-74 (1987).
48. L. M. Sandratskii, Phys. Status Solidi B 136(1), 167-180 (1986).
49. M. Pajda, J. Kudrnovsky, I. Turek, V. Drchal, and P. Bruno, Phys. Rev. B 64, 174402 (2001).
50. N. M. Rosengaard and B. Johansson, Phys. Rev. B 55, 14975 (1997).
51. M. Ležaić, P. Mavropoulos, and S. Blügel, Appl. Phys. Lett. 90, 082504 (2007).
52. A. V. Ruban, S. Khmelevskyi, P. Mohn, and B. Johansson, Phys. Rev. B 75, 054402 (2007).
53. J. Rusz, L. Bergqvist, J. Kudrnovský, and I. Turek, Phys. Rev. B 73, 214412 (2006).
54. G. Y. Gao, K. L. Yao, E. Şaşioǧlu, L. M. Sandratskii, Z. L. Liu, and J. L. Jiang, Phys. Rev. B 75, 174442 (2007).
55. F. Körmann, A. Dick, T. Hickel, and J. Neugebauer, Phys. Rev. B 79, 184406 (2009).
56. F. Körmann, A. Dick, T. Hickel, and J. Neugebauer, Phys. Rev. B 81(13), 134425 (2010).
57. W. Heisenberg, Z. Phys. 49(9-10), 619-636 (1928).
58. P. A. M. Dirac, Proc. R. Soc. Lond. A 118(779), 351-361 (1928).
59. M. Exler, On Classical and Quantum Mechanical Energy Spectra of Finite Heisenberg Spin Systems, Ph.D. thesis, University Osnabrück (2006).
60. P. W. Ma and S. L. Dudarev, Phys. Rev. B 86, 054416 (2012).
61. J. Oitmaa and W. Zheng, J. Phys.: Condens. Matter 16(47), 8653 (2004).
62. W. Nolting and A. Ramakanth, Quantum Theory of Magnetism (Springer, Berlin, 2008).
63. A. W. Sandvik and J. Kurkijärvi, Phys. Rev. B 43(7), 5950-5961 (1991).
64. H. G. Evertz, Adv. Phys. 1, 1 (2003).
65. P. Henelius and A. W. Sandvik, Phys. Rev. B 62(2), 1102-1113 (2000).
66. E. Y. Loh, J. E. Gubernatis, R. T. Scalettar, S. R. White, D. J. Scalapino, and R. L. Sugar, Phys. Rev. B 41(13), 9301-9307 (1990).
67. L. Bergqvist, B. Belhadji, S. Picozzi, and P. H. Dederichs, Phys. Rev. B 77(1), 014418 (2008).
68. F. Körmann, A. Dick, B. Grabowski, B. Hallstedt, T. Hickel, and J. Neugebauer, Phys. Rev. B 78, 033102 (2008).
69. F. Körmann, A. Dick, T. Hickel, and J. Neugebauer, Phys. Rev. B 83(16), 165114 (2011).
70. T. Hickel, B. Grabowski, F. Körmann, and J. Neugebauer, J. Phys.: Condens. Matter 24(5), 053202 (2012).
71. A. Dick, F. Körmann, T. Hickel, and J. Neugebauer, Phys. Rev. B 84, 125101 (2011).
72. R. Drautz and M. Fähnle, Phys. Rev. B 72, 212405 (2005).
73. R. Singer, F. Dietermann, and M. Fähnle, Phys. Rev. Lett. 107, 017204 (2011).
74. H. Capellmann, J. Magn. Magn. Mater. 28(3), 250-260 (1982).
75. M. Lavrentiev, S. Dudarev, and D. Nguyen-Manh, J. Nucl. Mater. 386-388, 22-25 (2009).
76. M. Y. Lavrentiev, D. Nguyen-Manh, and S. L. Dudarev, Phys. Rev. B 81, 184202 (2010).
77. P. M. Derlet, Phys. Rev. B 85, 174431 (2012).
78. P. W. Ma, C. H. Woo, and S. L. Dudarev, Phys. Rev. B 78(Jul), 024434 (2008).
79. P. W. Ma and C. H. Woo, Phys. Rev. E 79, 046703 (2009).
80. P. W. Ma, S. L. Dudarev, A. A. Semenov, and C. H. Woo, Phys. Rev. E 82, 031111 (2010).
81. P. W. Ma and S. L. Dudarev, Phys. Rev. B 83(Apr), 134418 (2011).
82. M. Y. Lavrentiev, R. Soulairol, C. C. Fu, D. Nguyen-Manh, and S. L. Dudarev, Phys. Rev. B 84, 144203 (2011).
83. J. Rogal et al., Phys. Status Solidi B, DOI 10.1002/pssb. 201350155 (2013).
84. M. J. Gillan, Contemp. Phys. 38(2), 115-130 (1997).
85. G. Ghosh, unpublished research, Northwestern University (2013).
86. M. Fallot, Ann. Phys. (France) 11e, 305 (1936).
87. C. Shull and M. K. Wilkinson, Phys. Rev. 97, 304 (1955).
88. M. Nevitt and A. Aldred, J. Appl. Phys. 34, 463 (1963).
89. A. Aldred, Phys. Rev. B 14, 219 (1976).
90. Y. Dorofeyev, A. Z. Menshikov, and G. Takzey, Phys. Met. Metallogr. 55, 102 (1983).
91. J. Blacktop, J. Crangle, and B. Argent, Mater. Sci. Technol. 1, 448 (1985).
92. P. Weiss and R. Ferrer, Ann. Phys. (France) 12, 279 (1929).
93. M. Collins and J. Forsyth, Philos. Mag. 8, 401 (1963).
94. D. Bardos, J. Appl. Phys. 40, 1371 (1969).
95. A. vande Walle and G. Ceder, J. Phase Equilib. 23(4), 348-359 (2002).
96. A. V. Ruban and I. Abrikosov, Rep. Prog. Phys. 71(4), 046501 (2008).
97. P. Olsson, I. Abrikosov, and J. Wallenius, Phys. Rev. B 73, 104416 (2006).
98. P. James, O. Eriksson, B. Johansson, and I. A. Abrikosov, Phys. Rev. B 59, 419-430 (1999).
99. G. Ghosh, A. vande Walle, and M. Asta, Acta Mater. 56, 3202 (2008).
100. G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169-11186 (1996).
101. P. E. Blöchl, Phys. Rev. B 50(24), 17953 (1994).
102. J. P. Perdew, J. Chevary, S. Vosko, K. A. Jackson, M. R. Pederson, D. Singh, and C. Fiolhais, Phys. Rev. B 46(11), 6671 (1992).
103. R. Richter and H. Eschrug, J. Phys. F, Met. Phys. 18, 1813 (1988).
104. M. van Schilfgaarde, I. Abrikosov, and B. Johansson, Nature 400(6739), 46-49 (1999).
105. E. Şaşioǧlu, L. M. Sandratskii, P. Bruno, and I. Galanakis, Phys. Rev. B 72(18), 184415 (2005).
106. J. Kübler, Theory of Itinerant Electron Magnetism (Clarendon, Oxford, 2000).
107. J. Kübler, Phys. Rev. B 67, 220403 (2003).
108. J. Kübler, J. Phys.: Condens. Matter 18, 9795 (2006).
109. J. Kübler, G. H. Fecher, and C. Felser, Phys. Rev. B 76, 024414 (2007).
110. B. Alling, A. V. Ruban, and I. A. Abrikosov, Phys. Rev. B 79, 134417 (2009).
111. B. Hallstedt, D. Djurovic, J. von Appen, R. Dronskowski, A. Dick, F. Körmann, T. Hickel, and J. Neugebauer, Calphad 34, 129 (2010).
112. P. Gustafson, Scand. J. Metallurgy 14(5), 259-267 (1985).
113. F. Bloch, Z. Phys. 61, 206 (1930).
114. R. Kubo, Phys. Rev. 87, 568 (1952).
115. O. Källbäck, S. G. Humble, and G. Malmström, Phys. Rev. B 24, 5214-5221 (1981).
116. B. Stroka, J. Wosnitza, E. Scheer, H. Löhneysen, W. Park, and K. Fischer, Z. Phys. B, Condens. Matter 89(1), 39-43 (1992).
117. S. N. Kaul and M. S. Rao, J. Phys.: Condens. Matter 6(36), 7403 (1994).
118. M. Marinelli, F. Mercuri, S. Foglietta, and D. P. Belanger, Phys. Rev. B 54, 4087-4092 (1996).
119. M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. B 65, 144520 (2002).
120. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford, 1987).
121. M. D. Kuz'min and A. M. Tishin, Europhys. Lett. 73(3), 396 (2006).
122. V. Franco, J. Blazquez, B. Ingale, and A. Conde, Annu. Rev. Mater. Res. 42, 305-42 (2012).
123. A. Arrott and J. Noakes, Phys. Rev. Lett. 19, 786-89 (1967).
124. M. D. Kuz'min, Phys. Rev. B 77, 184431 (2008).
125. M. D. Kuz'min, Phys. Rev. Lett. 94, 107204 (2005).
126. M. D. Kuz'min, M. Richter, and A. N. Yaresko, Phys. Rev. B 73, 100401 (2006).
127. L. D. Landau, Phys. Z. Sowjetunion 11, 26 (1937).
128. W. L. Bragg and E. J. Williams, Proc. R. Soc. Lond. A, Math. Phys. Sci. 145(855), 699-730 (1934).
129. W. L. Bragg and E. J. Williams, Proc. R. Soc. Lond. A, Math. Phys. Sci. 151(874), 540-566 (1935).
130. E. J. Williams, Proc. R. Soc. Lond. A, Math. Phys. Sci. 152(875), 231-252 (1935).
131. W. Gorsky, Z. Phys. 50(1-2), 64-81 (1928).
132. W. Shockley, J. Chem. Phys. 6(3), 130-144 (1938).
133. I. Ansara, B. Sundman, and P. Willemin, Acta Metall. 36(4), 977-982 (1988).
134. J. Lacaze and B. Sundman, Metall. Trans. A 22(10), 2211-2223 (1991).
135. N. Dupin and B. Sundman, Scand. J. Metallurgy 30(3), 184-192 (2001).
136. R. Kikuchi, Phys. Rev. 81, 988 (1951).
137. R. Kikuchi and S. G. Brush, J. Chem. Phys. 47, 195 (1967).
138. J. A. Barker, Proc. R. Soc. A216, 45 (1953).
139. J. Hijmans and J. De Boer, Physica 21, 471 (1955).
140. T. Morita, J. Phys. Soc. Jpn. 12, 753 (1957).
141. T. Morita, J. Math. Phys. 13, 115 (1972).
142. R. Kikuchi and J. Murray, Calphad 9(4), 311-348 (1985).
143. C. G. Schön and G. Inden, Comput. Mater. Sci. 20(1), 98-106 (2001).
144. J. W. D. Connolly and A. R. Williams, Phys. Rev. B 27, 5169 (1983).
145. J. M. Sanchez, F. Ducastelle, and D. Gratias, Physica A 128, 334 (1984).
146. T. Mohri, J. Sanchez, and D. De Fontaine, Acta Metall. 33(7), 1171-1185 (1985).
147. A. Kusoffsky, N. Dupin, and B. Sundman, Calphad 25(4), 549-565 (2001).
148. B. Sundman, S. G. Fries, and W. Oates, Calphad 22(3), 335-354 (1998).
149. B. Sundman, I. Ohnuma, N. Dupin, U. R. Kattner, and S. G. Fries, Acta Mater. 57(10), 2896-2908 (2009).
150. R. Castanet and A. Ferrier, Compt. Rend. 15, 272C (1971).
151. I. Ohnuma, H. Enoki, O. Ikeda, R. Kainuma, H. Ohtani, B. Sundman, and K. Ishida, Acta Mater. 50, 379 (2002).
152. I. Ohnuma, S. Abe, S. Shimenouchi, T. Omori, R. Kainuma, and K. Ishida, ISIJ Int. 52(4), 540-548 (2012).
153. M. Cadeville and J. Moran-Lopez, Phys. Rep. 153(6), 33-399 (1987).
154. A. Berkowitz and E. Kneller, Magnetism and Metallurgy (Academic Press Inc., New York, 1969).
155. J. B. Staunton, D. D. Johnson, and B. L. Gyorffy, J. Appl. Phys. 61(8), 3693-3696 (1987).
156. A. J. Pindor, J. Staunton, G. M. Stocks, and H. Winter, J. Phys. F, Met. Phys. 13(5), 979 (1983).
157. A. V. Ruban, S. Shallcross, S. Simak, and H. L. Skriver, Phys. Rev. B 70(12), 125115 (2004).
158. M. Rahaman, A. Ruban, A. Mookerjee, and B. Johansson, Phys. Rev. B 83(5), 054202 (2011).
159. M. Ekholm, H. Zapolsky, A. Ruban, I. Vernyhora, D. Ledue, and I. Abrikosov, Phys. Rev. Lett. 105(16), 167208 (2010).
160. J. M. Sanchez, M. C. Cadeville, V. Pierron-Bohnes, and G. Inden, Phys. Rev. B 54, 8958 (1996).
161. J. I. Seki, M. Hagiwara, and T. Suzuki, J. Mater. Sci. 14, 2404 (1979).
162. Y. Kakehashi, J. Magn. Magn. Mater. 54-57, 1096 (1986).
163. M. V. Nevitt, J. Appl. Phys. 31, 155 (1960).
164. I. Mirebeau, G. Parette, and J. W. Cable, J. Phys. F, Met. Phys. 17, 191 (1987).
165. D. D. Johnson, F. J. Pinski, and J. B. Staunton, J. Appl. Phys. 61, 3715 (1987).
166. J. C. Krause, J. Schaff, M. I. Costa, and C. Paduani, Phys. Rev. B 61, 6196 (2000).
167. J. A. Muñoz, M. S. Lucas, O. Delaire, M. L. Winterrose, L. Mauger, C. W. Li, A. O. Sheets, M. B. Stone, D. L. Abernathy, Y. Xiao, P. Chow, and B. Fultz, Phys. Rev. Lett. 107, 115501 (2011).
168. R. G. Jordan, X. Wang, A. M. Begley, S. L. Qiu, and Y. Liu, Solid State Commun. 78, 1045 (1991).
169. V. L. Moruzzi and P. M. Marcus, Phys. Rev. B 45, 2934 (1992).
170. M. S. Lucas, J. A. Muñoz, O. Delaire, N. Markovskiy, M. B. Stone, D. Abernathy, I. Halevy, L. Mauger, B. Keith, M. L. Winterrose, Y. Xiao, M. Lerche, and B. Fultz, Phys. Rev. B 82, 144306 (2010).
171. J. O. Andersson, Calphad 7, 305 (1983).
172. R. Wakelin and E. Yates, Proc. Phys. Soc. Lond. B 66, 221 (1953).
173. I. Ohnuma, O. Ikeda, R. Kainuma, B. Sundman, and K. Ishida, Z. Metallkd. 89, 847 (1998).
174. I. Ohnuma, R. Kainuma, and K. Ishida, in: CALPHAD and Alloy Thermodynamics, Proc. of a Symposium held at the 2002 TMS Annual Meeting, Seattle, WA, February 2002 (TMS, Warrendale, PA, 2002), pp. 61-78.
175. I. Ohnuma et al., Paper presented at the CALPHAD XXVII, Beijing, China, May 18, 1998.
176. O. Ikeda, I. Ohnuma, R. Kainuma, and K. Ishida, Intermetallics 9, 755 (2001).
177. L. Kaufman, E. V. Clougherty, and R. J. Weiss, Acta Metall. 11, 323 (1963).
178. H. Hasegawa and D. G. Pettifor, Phys. Rev. Lett. 50, 130-133 (1983).
179. I. Leonov, A. I. Poteryaev, V. I. Anisimov, and D. Vollhardt, Phys. Rev. Lett. 106, 106405 (2011).
180. J. Zarestky and C. Stassis, Phys. Rev. B 35, 4500-4502 (1987).
181. H. C. Hsueh, J. Crain, G. Y. Guo, H. Y. Chen, C. C. Lee, K. P. Chang, and H. L. Shih, Phys. Rev. B 66, 052420 (2002).
182. F. Körmann, Magnetic systems studied by first-principles thermodynamics, Ph.D. thesis, University of Paderborn (2011), http://digital.ub.uni-paderborn.de/hs/download/pdf/ 8865.
183. H. Hasegawa, M. W. Finnis, and D. G. Pettifor, J. Phys. F, Met. Phys. 17(10), 2049 (1987).
184. I. Leonov, A. I. Poteryaev, V. I. Anisimov, and D. Vollhardt, Phys. Rev. B 85, 020401 (2012).
185. A. V. Ruban and V. I. Razumovskiy, Phys. Rev. B 85, 174407 (2012).
186. F. Körmann, A. Dick, B. Grabowski, T. Hickel, and J. Neugebauer, Phys. Rev. B 85, 125104 (2012).
187. D. Dever, J. Appl. Phys. 43(8), 3293-3301 (1972).
188. S. L. Dudarev, R. Bullough, and P. M. Derlet, Phys. Rev. Lett. 100, 135503 (2008).
189. Z. Yao, M. L. Jenkins, M. Hernandez-Mayoral, and M. A. Kirk, Philos. Mag. 90(35-36), 4623-4634 (2010).
190. E. Martinez, O. Senninger, C. C. Fu, and F. Soisson, Phys. Rev. B 86, 224109 (2012).
191. M. Y. Lavrentiev, K. Mergia, M. Gjoka, D. Nguyen-Manh, G. Apostolopoulos, and S. L. Dudarev, J. Phys.: Condens. Matter 24(32), 326001 (2012).
192. M. Y. Lavrentiev, S. L. Dudarev, and D. Nguyen-Manh, J. Appl. Phys. 109(7), 07E123 (2011).