Phase stability and thermo-physical properties of Laves-Co<sub>2</sub>(Hf Ta) alloys

Ding Ning; Ni Xiao-Dong; Tian Fu-Yang

doi:10.7498/aps.73.20240143

Abstract

There exists still the controversy over the stable structure of Laves-phase Co₂(Hf Ta) alloys with the C14, C15 or C36 structures. In this study, the stability, electronic and thermodynamic properties of Laves-phase Co₂(Hf Ta) are investigated. In order to fully understand the influence of magnetic state and temperature on phase stability, we systematically study the free energy change at finite temperature, elastic stability, and phonon dispersion. The low Curie temperature can be estimated, which suggests that the Co₂(Hf Ta) alloys possess the paramagnetic state in a wide temperature range. Results indicate that the lattice vibration and electronic excitation have an important effect on the phase stability. The ground state of Co₂Hf compound has a C14-type structure, while the ground state of Co₂Ta has a C36-type structure, without the effect of temperature. After doping Hf with different concentrations (0.25, 0.50, 0.75) into Co₂Ta, the most stable structure still possesses the C36-type structure. After considering the contribution of vibration entropy and electron entropy, the relatively stable structures of Co₂Hf and Co₂Ta undergo the C36 and C14 phase transition, respectively. In addition, the thermodynamic properties, including Debye temperature, heat capacity, and vibration entropy, which vary with pressure and temperature, are studied. The electronic properties of Co₂Hf and Co₂Ta compounds are analyzed by the charge difference and density of states. The similar electronic density of states between different phases suggest that the Lave phases have the similar stability. The Hf-Co bonding with a certain direction is revealed. Our results are of great significance in understanding the structure and properties of Co₂Hf and Co₂Ta compounds.

Keywords:

Authors and contacts

Institute for Applied Physics, University of Science and Technology Beijing, Beijing 100083, China

Corresponding author: Ni Xiao-Dong, nixd@ustb.edu.cn ; Tian Fu-Yang, fuyang@ustb.edu.cn

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References

[1]	Schulze G E 1939 Zeitschrift Für Elektrochemie und Angewandte Physikalische Chemie 45 849 Google Scholar
[2]	Stein F, Leineweber A 2021 J. Mater. Sci. 56 5321 Google Scholar
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[4]	Zhang Y, Fu H, He J, Xie J 2022 J. Alloys Compd. 891 162016 Google Scholar
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[6]	Chen K C, Peterson E J, Thoma D J 2001 Intermetallics 9 771 Google Scholar
[7]	Chen K C, Chu F, Kotula P G, Thoma D 2001 Intermetallics 9 785 Google Scholar
[8]	Concas G, Congiu F, Belošević-Čavor J, Cekić B 2007 Zeitschrift für Naturforschung A 62 452 Google Scholar
[9]	Wang C, Chen X, Yang P, Zhang Q, Yang S, Lu Y, Guo Y, Liu X 2022 J. Alloys Compd. 925 166723 Google Scholar
[10]	Ooshima M, Tanaka K, Okamoto N L, Kishida K, Inui H 2010 J. Alloys Compd. 508 71 Google Scholar
[11]	Omori T, Oikawa K, Sato J, Ohnuma I, Kattner U R, Kainuma R, Ishida K 2013 Intermetallics 32 274 Google Scholar
[12]	Povstugar I, Choi P P, Neumeier S, Bauer A, Zenk C H, Göken M, Raabe D 2014 Acta Mater. 78 78 Google Scholar
[13]	Dragsdorf R, Foreing W 1962 Acta Crystallogr. 15 531 Google Scholar
[14]	Baheti V A, Roy S, Ravi R, Paul A 2013 Intermetallics 33 87 Google Scholar
[15]	Shinagawa K, Chinen H, Omori T, Oikawa K, Ohnuma I, Ishida K, Kainuma R 2014 Intermetallics 49 87 Google Scholar
[16]	Wang P, Koßmann J, Kattner U R, Palumbo M, Hammerschmidt T, Olson G B 2019 Calphad 64 205 Google Scholar
[17]	Zhou C, Guo C, Li C, Du Z 2019 Calphad 66 101649 Google Scholar
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[19]	Stein F, Palm M, Sauthoff G 2004 Intermetallics 12 713 Google Scholar
[20]	Von Keitz A, Sauthoff G 2002 Intermetallics 10 497 Google Scholar
[21]	Thoma D, Perepezko J 1995 J. Alloys Compd. 224 330 Google Scholar
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Cited By

图 1 Laves相三种类型结构示意图　(a) C14结构; (b) C15结构; (c) C36结构

Figure 1. Schematic diagrams of three types of Laves phase: (a) C14 structure; (b) C15 structure; (c) C36 structure.

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图 2 在0 GPa下Co₂Hf (a) 和Co₂Ta (b) Laves相的FM态和PM态的声子色散曲线

Figure 2. Phonon dispersion of Co₂Hf (a) and Co₂Ta (b) Laves phases with FM and PM at 0 GPa.

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图 3 Co₂(Hf_xTa_1–x)合金的晶格参数(a)和形成能(b)随Hf含量x的变化

Figure 3. Lattice parameters (a) and formation energy (b) of Co₂(Hf_xTa_1–x) alloys versus the Hf composition.

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图 4 Co₂Hf (a)和 Co₂Ta (b)的弹性模量、泊松比($ \nu $)及B/G参数

Figure 4. Elastic modulus, Poisson's ratio ($ \nu $) and B/G for Co₂Hf (a) and Co₂Ta (b).

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图 5 Co₂(Hf Ta)合金的弹性模量、泊松比($ \nu $)和及B/G参数

Figure 5. Elastic modulus, Poisson’s ratio ($ \nu $) and B/G values of Co₂(Hf Ta) alloys.

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图 6 Co₂Hf (a)和Co₂Ta (b) 的弹性模量和各向异性因子A^U, A_B和A_G

Figure 6. Elastic moduli and anisotropy factors A^U, A_B and A_G of Co₂Hf (a) and Co₂Ta (b).

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图 7 Co₂Ta_1–xHf_x稳定结构的杨氏模量 (E)的三维图(颜色刻度值以GPa为单位)　(a) 铁磁态C14相Co₂Hf; (b) 顺磁态C15相Co₂Hf; (c) 铁磁态C36相 Co₂Ta; (d)—(f) 铁磁C36相 Co₂Hf_0.25Ta_0.75, Co₂Hf_0.5Ta_0.5, Co₂Hf_0.75Ta_0.25

Figure 7. Anisotropy of Young’s modulus (E) of Co₂Ta_1–xHf_x along different crystalline directions: (a) Ferromagnetic C14 Co₂Hf; (b) paramagnetic C15 Co₂Hf; (c) ferromagnetic C36 Co₂Ta; (d)–(f) ferromagnetic C36 Co₂Hf_0.25Ta_0.75, Co₂Hf_0.5Ta_0.5 and Co₂Hf_0.75Ta_0.25. The color scale is in GPa.

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图 8 不同压强下热容随温度的变化　(a) C14-PM Co₂Hf; (b) C36-PM Co₂Ta; (c) C36- PM Co₂Hf_0.25Ta_0.75; (d) C36-PM Co₂Hf_0.5Ta_0.5; (e) C36-PM Co₂Hf_0.75Ta_0.25

Figure 8. Heat capacity with temperature under different pressures: (a) C14-PM Co₂Hf; (b) C36-PM Co₂Ta; (c) C36-PM Co₂Hf_0.25Ta_0.75; (d) C36-PM Co₂Hf_0.5Ta_0.5; (e) C36-PM Co₂Hf_0.75Ta_0.25.

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图 9 德拜温度与温度和压强的关系　(a) C15 Co₂Hf; (b) C36 Co₂Ta; (c) C36 Co₂Hf_0.25Ta_0.75; (d) C36 Co₂Hf_0.5Ta_0.5; (e) C36 Co₂Hf_0.75Ta_0.25

Figure 9. Debye temperature versus temperature at finite pressure: (a) C15 Co₂Hf; (b) C36 Co₂Ta; (c) C36 Co₂Hf_0.25Ta_0.75; (d) C36 Co₂Hf_0.5Ta_0.5; (e) C36 Co₂Hf_0.75Ta_0.25.

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图 10 不同压强下, 振动熵与温度的变化关系, 其中(a) C15 Co₂Hf; (b) C36 Co₂Ta; (c)—(e) C36 Co₂Hf_0.25Ta_0.75, Co₂Hf_0.5Ta_0.5, Co₂Hf_0.75Ta_0.25; (f) Co₂Hf和Co₂Ta 的自由能随温度的变化

Figure 10. Vibrational entropy versus temperature under different pressures: (a) C15 Co₂Hf; (b) C36 Co₂Ta; (c)–(e) C36 Co₂Hf_0.25Ta_0.75, Co₂Hf_0.5Ta_0.5, Co₂Hf_0.75Ta_0.25. (f) The free energy versus temperature for Co₂Hf and Co₂Ta.

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图 11 Co₂Hf和Co₂Ta总态密度和分态密度(垂直虚线表示费米能级)　(a), (b) C14, C15, C36相的Co₂Hf和 Co₂Ta的总电子态密度; (c), (e) C14相 Co₂Hf的分态密度; (d), (f) C36相Co₂Ta的分态密度

Figure 11. Total density of states and partial density of states of Co₂Hf and Co₂Ta : (a), (b) The total electronic density of states of Co₂Hf and Co₂Ta in C14, C15 and C36 phases; (c), (e) the partial density of states of C14 phase Co₂Hf; (d), (f) the partial density of states of C36 phase Co₂Ta. The vertical dotted line represents the Fermi level.

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图 12 差分电荷密度图　(a) Co₂Hf 的C14结构$ (11 \bar{2} 0)$面; (b) Co₂Hf 的C15结构的(110)面; (c), (d) 代表为铁磁和顺磁Co₂Ta的C36 的$(11 \bar{2}0) $面

Figure 12. Differential charge density diagram: (a) C14-FM $ (11\bar{2}0) $ plane of C14 structure of Co₂Hf; (b) C15-FM (110) plane of C15 structure of Co₂Hf; (c), (d) the $ (11\bar{2} 0)$ plane of C36 with ferromagnetic and paramagnetic Co₂Ta.

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表 1 Laves相晶体结构的基本信息; 表中x = 1/6, z (C14-A₁) = 9/16, z (C36-A₁) = 21/32, z (C36-A₂) = 3/32, z (C36-B₃) = 1/8理想原子位置附近, 随A和B的不同, 6h和4e, 4f位的位置参数x和z会有微小的变化^[2]

Table 1. Details on Laves phase crystal structure. Depending on A and B sites, the position parameters x and z of the 6h and 4e, 4f sites slightly vary around the ideal atom position values of x = 1/6, z (C14-A₁) = 9/16, z (C36-A₁) = 21/32, z (C36-A₂) = 3/32, z (C36-B₃) = 1/8^[2].

结构名称	结构类型	空间群	原子	亚格点位置以及占位
结构名称	结构类型	空间群	原子		x	y	z
C15	MgCu₂	227	A	8a	0	0	0
C15	MgCu₂	227	B	16d	5/8	5/8	5/8
C14	MgZn₂	194	A	4f	1/3	2/3	z
			B₁	6h	x	2x	1/4
			B₂	2a	0	0	0
C36	MgNi₂	194	A₁	4f	1/3	2/3	z
			A₂	4e	0	0	z
			B₁	6h	x	2x	1/4
			B₂	6g	1/2	0	0
			B₃	4f	1/3	2/3	z

DownLoad: CSV

表 2 Co₂Hf和Co₂Ta Laves相的晶格常数a和c、体积V、钴的局域磁矩M、形成能ΔH、有无自旋极化状态下的能量差ΔE, 及可利用的实验或模拟数据

Table 2. Lattice constants a, c, volume V, local magnetic moment M, formation energy ΔH, energy difference ΔE with and without spin polarization state , and available experimental or simulated data of Co₂Hf and Co₂Ta Laves phases.

体系	结构	a/Å	c/Å	V/(Å³·atom^–1)	M/(μ_B·atom^–1)	ΔH/(eV·atom^–1)	ΔE/(eV·atom^–1)	Refs.
Co₂Hf	C14	4.873	7.939	13.605	0.69 0.79	–0.3788	–0.017
	C15	4.870		13.615	0.76	–0.3752	–0.002
	C15	4.824 4.909		13.945				Cal.^[22] Cal.^[34]
	C36	4.876	15.837	13.585	0.68 0.70 0.58	–0.3779	–0.007
	C36	4.816	15.650					Cal.^[22]
Co₂Ta	C14	4.766	7.750	12.703	0.33 0.68	–0.2378	–0.002
	C14	4.797	7.827	12.730				Exp.^[13]
	C15	4.759		12.702	0.40	–0.2442	–0.005
	C15	4.791		12.957				Cal.^[34]
	C36	4.760	15.529	12.698	0.39 0.41 0.27	–0.2447	–0.003

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表 3 Co₂Hf和Co₂Ta的C14, C15和C36结构以及Co₂(Ta_1–xHf_x)基态结构的弹性常数c_ij (GPa), 其中, 结构部分, FM和PM代表铁磁和顺磁态, 数字156, 186表示空间群

Table 3. Elastic constants c_ij (GPa) of C14, C15, C36Co₂Hf and Co₂Ta as well as Co₂(Ta_1–xHf_x), where the structural part, FM and PM represent ferromagnetic and paramagnetic states, and the numbers 156, 186 represent the space group.

体系	结构	c₁₁	c₃₃	c₄₄	c₁₂	c₁₃	$c_{66}, {c_{14}}^* $	Refs.
Co₂Hf	C14-FM	274.14	332.76	81.86	86.14	126.48	94.00
	C14-PM	299.75	348.07	73.80	90.11	113.04	104.82
	C15-FM	308.80		85.60	154.53
	C15-PM	298.50		106.00	139.12
	C15	270.20		85.50	117.50			Cal.^[34]
	C36-FM	308.11	293.50	80.53	121.99	113.90	93.06
	C36-PM	329.08	343.58	81.90	120.83	117.89	104.12
Co₂Ta	C14-FM	407.15	430.80	97.10	138.20	151.11	134.48
	C14-PM	442.04	421.53	89.27	167.32	137.34	137.36
	C15-FM	357.11		141.47	137.90
	C15-PM	395.37		151.94	165.32
	C15	383.50		143.90	173.40			Cal.^[34]
	C36-FM	424.64	429.32	98.18	154.27	143.77	135.19
	C36-PM	444.08	481.10	88.70	174.90	177.34	134.59
Co₂Hf_0.25Ta_0.75	156-FM	390.67	383.02	80.43	153.72	146.41	–1.13
Co₂Hf_0.5Ta_0.5	186-FM	358.36	362.80	84.35	132.40	126.78
Co₂Hf_0.75Ta_0.25	156-FM	327.16	312.04	77.66	130.90	122.88	0.68
*: c₆₆, c₁₄数据列, 其中后3行时c₁₄数值, 其他行为c₆₆数值.

DownLoad: CSV

[1]	Schulze G E 1939 Zeitschrift Für Elektrochemie und Angewandte Physikalische Chemie 45 849 Google Scholar
[2]	Stein F, Leineweber A 2021 J. Mater. Sci. 56 5321 Google Scholar
[3]	张硕鑫, 刘士余, 严达利, 余浅, 任海涛, 于彬, 李德军 2021 物理学报 70 117102 Google Scholar Zhang S X, Liu S Y, Yan D L, Yu Q, Ren H T, Yu B, Li D J 2021 Acta Phys. Sin. 70 117102 Google Scholar
[4]	Zhang Y, Fu H, He J, Xie J 2022 J. Alloys Compd. 891 162016 Google Scholar
[5]	Burzo E, Gratz E, Pop V 1993 J. Magn. Magn. Mater. 123 159 Google Scholar
[6]	Chen K C, Peterson E J, Thoma D J 2001 Intermetallics 9 771 Google Scholar
[7]	Chen K C, Chu F, Kotula P G, Thoma D 2001 Intermetallics 9 785 Google Scholar
[8]	Concas G, Congiu F, Belošević-Čavor J, Cekić B 2007 Zeitschrift für Naturforschung A 62 452 Google Scholar
[9]	Wang C, Chen X, Yang P, Zhang Q, Yang S, Lu Y, Guo Y, Liu X 2022 J. Alloys Compd. 925 166723 Google Scholar
[10]	Ooshima M, Tanaka K, Okamoto N L, Kishida K, Inui H 2010 J. Alloys Compd. 508 71 Google Scholar
[11]	Omori T, Oikawa K, Sato J, Ohnuma I, Kattner U R, Kainuma R, Ishida K 2013 Intermetallics 32 274 Google Scholar
[12]	Povstugar I, Choi P P, Neumeier S, Bauer A, Zenk C H, Göken M, Raabe D 2014 Acta Mater. 78 78 Google Scholar
[13]	Dragsdorf R, Foreing W 1962 Acta Crystallogr. 15 531 Google Scholar
[14]	Baheti V A, Roy S, Ravi R, Paul A 2013 Intermetallics 33 87 Google Scholar
[15]	Shinagawa K, Chinen H, Omori T, Oikawa K, Ohnuma I, Ishida K, Kainuma R 2014 Intermetallics 49 87 Google Scholar
[16]	Wang P, Koßmann J, Kattner U R, Palumbo M, Hammerschmidt T, Olson G B 2019 Calphad 64 205 Google Scholar
[17]	Zhou C, Guo C, Li C, Du Z 2019 Calphad 66 101649 Google Scholar
[18]	Asano S, Ishida S 1988 J. Phys. F: Metal Phys. 18 501 Google Scholar
[19]	Stein F, Palm M, Sauthoff G 2004 Intermetallics 12 713 Google Scholar
[20]	Von Keitz A, Sauthoff G 2002 Intermetallics 10 497 Google Scholar
[21]	Thoma D, Perepezko J 1995 J. Alloys Compd. 224 330 Google Scholar
[22]	Wang L, Kong Y, Oehring M, Song M, Pyczak F 2022 J. Alloys Compd. 906 164261 Google Scholar
[23]	Levy O, Hart G L, Curtarolo S 2010 Acta Mater. 58 2887 Google Scholar
[24]	Segall M, Lindan P J, Probert M A, Pickard C J, Hasnip P J, Clark S, Payne M 2002 J. Phys.: Condens. Matter 14 2717 Google Scholar
[25]	Clark S J, Segall M D, Pickard C J, Hasnip P J, Probert M I, Refson K, Payne M C 2005 Z. Krist. -Cryst. Mater. 220 567 Google Scholar
[26]	Perdew J P, Ruzsinszky A, Csonka G I, Vydrov O A, Scuseria G E, Constantin L A, Zhou X, Burke K 2008 Phys. Rev. Lett. 100 136406 Google Scholar
[27]	Monkhorst H J, Pack J D 1976 Phys. Rev. B 13 5188 Google Scholar
[28]	Pfrommer B G, Côté M, Louie S G, Cohen M L 1997 J. Comput. Phys. 131 233 Google Scholar
[29]	Wang Q, Li Q, Ding H, Tian F 2022 Comput. Condens. Matter 32 e00721 Google Scholar
[30]	Sun Y, Zhou Y, Gurunathan R, Zhang J Y, Hu M, Liu W, Xu B, Snyder G J 2021 J. Mater. Chem. C 9 8506 Google Scholar
[31]	Wu R, Wang Y P, Shao L, Wang W, Tang B Y 2021 Chin. J. Chem. Eng. 40 315 Google Scholar
[32]	Gao J, Liu Q, Jiang C, Fan D, Zhang M, Liu F, Tang B 2022 Chin. J. High Press. Phys 36 051101 Google Scholar
[33]	Jana A, Sridar S, Fries S G, Hammerschmidt T, Kumar K H 2020 Intermetallics 116 106640 Google Scholar
[34]	Chen S, Sun Y, Duan Y H, Huang B, Peng M J 2015 J. Alloys Compd. 630 202 Google Scholar
[35]	Özer T 2020 Canadian J. Phys. 98 357 Google Scholar
[36]	Cheng T M, Yu G L, Zhang X X 2022 Physica B 645 414268 Google Scholar
[37]	Xu Y, Cao Y, Hu C, Zhang C, Tian C, Alzoabi S, Santos N, Zhou S 2021 Solid State Sci. 115 106580 Google Scholar
[38]	Xu N, Chen Y, Chen S, Zhang W, Li S, Song R, Zhang J 2023 J. Mater. Res. Technol. 26 4825 Google Scholar
[39]	Jian Y, Huang Z, Xing J, Sun L, Liu Y, Gao P 2019 Mater. Chem. Phys. 221 311 Google Scholar
[40]	Ranganathan S I, Ostoja-Starzewski M 2008 Phys. Rev. Lett. 101 055504 Google Scholar
[41]	Hong D, Zeng W, Liu F S, Tang B, Liu Q J 2019 Physica B 558 100 Google Scholar

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Vol. 73, Issue 12 - 2024-06-20

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Phase stability and thermo-physical properties of Laves-Co₂(Hf Ta) alloys

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Corresponding author: Ni Xiao-Dong, nixd@ustb.edu.cn ; Tian Fu-Yang, fuyang@ustb.edu.cn

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Phase stability and thermo-physical properties of Laves-Co2(Hf Ta) alloys

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Corresponding author: Ni Xiao-Dong, nixd@ustb.edu.cn ; Tian Fu-Yang, fuyang@ustb.edu.cn

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Phase stability and thermo-physical properties of Laves-Co₂(Hf Ta) alloys