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运用基于密度泛函理论的第一性原理方法, 对Co2FeAl1–xSix(x = 0.25, 0.5, 0.75)系列Heusler合金的电子结构、四方畸变、弹性常数, 声子谱以及热电特性进行了计算研究. 结果显示, Co2FeAl1–xSix系列合金的电子结构均为半金属特性, 向下自旋态(半导体性)均呈现良好的热电特性, 并且随着硅原子浓度的增加功率因子随之增加. 计算的声子谱不存在虚频, 均满足动力学稳定性条件, 弹性常数均满足玻恩稳定性条件, 机械稳定性均良好. 随着晶格常数c/a的比值变化, 体系的能量最低点均出现在c/a = 1处, 即结构稳定性不随畸变度c/a的变化而变化, 说明不存在马氏体相变. 此系列合金薄膜的电子结构呈现较高的自旋极化率, 在替代浓度x = 0.75时自旋极化率达到100%, 且当x = 0.75时薄膜在畸变度c/a = 1.2时存在马氏体相变. 随着晶格畸变度的改变, 总磁矩也发生变化, 且主要由Fe和Co两种过渡金属原子的磁矩变化所决定.In the recent decades, the half-metallic materials have become a research hotspot because of their unique electronic structure. The 100% spin polarization at the Fermi level makes them widely used in spintronic devices. The Co-based Heusler alloys belong to an important class of magnetic material, and Co2FeAl and Co2FeSi have been experimentally confirmed to be half-metallic materials with 100% spin polarization at the Fermi level, and the Co2FeSi has a high Curie temperature of 1100 K and a large magnetic moment of 6.0
${{\text{μ}}{\rm{B}}}$ , which is a good candidate for spintronic devices. We here choose and substitute Al atoms in Co2FeAl with Si atoms, and then carry out the theoretical predictions of Co2FeAl1–xSix (x = 0.25, 0.5, 0.75) for both bulk and film . In this paper, using the first principles calculations based on the density functional theory (DFT) we study the electronic structure, tetragonal distortion, elastic constants, phonon spectrum and thermoelectric properties of Co2FeAl1–xSix (x = 0.25, 0.5, 0.75) series alloys. The calculation results show that the electronic structure of Co2FeAl1–xSix (x = 0.25, 0.5, 0.75) series alloys are all half-metallic with 100% spin polarization, and the down spin states (semiconducting character) all exhibit good thermoelectric properties, and the power factor increases with the substitution concentration of Si atoms increasing. The calculated phonon spectrum does not have virtual frequency, indicating its dynamic stability, and all cubic phases fulfill the mechanical stability criteria, i.e. Born criteria: C11 > 0, C44 > 0, C11–C12 > 0, C11 + 2C12 > 0, and C12 < B < C11. With the variation of lattice constant ratio c/a, the lowest energy point of the structure for Co2FeAl1–xSix (x = 0.25, 0.5, 0.75) series alloys are all at c/a = 1, showing that the stability of the structure does not change with the variation of distortion c/a, and further the martensitic transformation cannot occur. For the Co2FeAl1–xSix (x = 0.25, 0.5, 0.75) series alloy thin films, the calculated electronic structures all show a high spin polarization, and it reaches 100% at x = 0.75, and for x = 0.75, the lowest energy point of the structure is at c/a = 1.2, suggesting the martensitic transformation in this structure. With the variation of the tetragonal distortion, the total magnetic moment also changes and it is mainly determined by the changes of atomic magnetic moment of transition-metals Fe and Co.-
Keywords:
- half-metallic /
- first principles /
- electronic structure /
- magnetism
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图 1 (a) Co2FeAl1－xSix (x = 0.25)的L21结构; (b) Co2FeAl0.75Si0.25的薄膜结构
Fig. 1. (a) L21 structure of Co2FeAl1－xSix (x = 0.25); (b)thin film structure of Co2FeAl0.75Si0.25.
图 2 Co2FeAl1－xSix合金在铁磁态(FM)和反铁磁态(AFM)下的晶格常数优化曲线 (a) x = 0.25; (b) x = 0.5; (c) x = 0.75
Fig. 2. Optimization curves of lattice constant for Co2FeAl1－xSix alloy under ferromagnetic and antiferromagnetic magnetic order.
图 3 (a) Co2FeAl0.75Si0.25, (b) Co2FeAl0.5Si0.5和(c) Co2FeAl0.25Si0.75的能带结构
Fig. 3. Energy band structure of (a) Co2FeAl0.75Si0.25, (b) Co2FeAl0.5Si0.5 and (c) Co2FeAl0.25Si0.75.
图 4 (a) Co2FeAl0.75Si0.25,(b) Co2FeAl0.5Si0.5和(c) Co2FeAl0.25Si0.75的总态密度和分态密度
Fig. 4. Thetotaland atom-projected density of states for Heusler alloys Co2FeAl1－xSix (x = 0.25, 0.5, 0.75） film in (a), (b) and (c).
图 5 Co2FeAl0.75Si0.25向下自旋态的(a)Seebeck系数, (b)电导, (c)热导和(d)功率因子随化学势的变化; Co2FeAl0.5Si0.5向下自旋态的(e) Seebeck系数, (f)电导, (g)热导和(h) 功率因子随化学势的变化; Co2FeAl0.25Si0.75向下自旋态的(i)Seebeck系数, (j)电导, (k)热导和(l)功率因子随化学势的变化
Fig. 5. The transport properties with variation of chemical potential
$\mu $ for Co2FeAl1－xSix(x = 0.25, 0.5 and 0.75). The case of x = 0.25 corresponds to (a), (b), (c) and (d), and the case of x = 0.5 corresponds to (e), (f), (g) and (h), and the case of x = 0.75 corresponds to (i), (j), (k) and (l). The four columns from left to right correspond to the Seebeck coefficients S, electrical conductivity$\sigma $ , electronic thermal conductivity${\kappa _{\rm{e}}}$ and PF (${S^2}\sigma $ ), respectively.
图 6 Co2FeAl1－xSix合金在x = 0.25, 0.5, 0.75时的声子谱及比热容 (a) Co2FeAl1－xSix (x = 0.25), (b) Co2FeAl1－xSi x (x = 0.5)和(c) Co2FeAl1－xSi x (x = 0.75)的声子谱; (d) Co2FeAl1－ xSix (x = 0.25, 0.5, 0.75)的比热容随温度的变化
Fig. 6. Full phonon spectra of Co2FeAl1－xSix (x = 0.25, 0.5 and 0.75) alloys in (a), (b) and (c). The temperature dependent heat capacity Cv with an inset graph showing the temperaturefrom 180 K to 250 K in (d).
图 7 (a) Co2FeAl0.75Si0.25, (b) Co2FeAl0.5Si0.5和(c) Co2FeAl0.25Si0.75薄膜的总态密度和原子分态密度
Fig. 7. Thetotaland atom-projected density of states for Co2FeAl1－xSix (x = 0.25, 0.5 and 0.75) film in (a), (b) and (c).
图 8 (a) x = 0.25,(b) x = 0.5和(c) x = 0.75替代浓度下Co2FeAl1－xSix合金体相的总能量差
$\Delta E$ 与畸变度c/a的关系; (d) x = 0.25, (e) x = 0.5和(f) x = 0.75替代浓度下Co2FeAl1－xSix薄膜的驱动力$\Delta E$ 与畸变度c/a的关系Fig. 8. Calculated total energies as a function of the c/a ratio for Co2FeAl1－xSix (x = 0.25, 0.5 and 0.75) Heusler alloys in (a), (b) and (c) andfilm materials in (d), (e) and (f).
图 9 (a) x = 0.25,(b) x = 0.5和(c) x = 0.75替代浓度下Co2FeAl1－xSix合金薄膜的总磁矩及各原子总磁矩随畸变度的变化
Fig. 9. The total magnetic moment and the magnetic moment of each atom of Co2FeAl1－xSix film change with distortion at x = 0.25, x = 0.5 and x = 0.75 in (a), (b) and (c).
表 1 Co2FeAl1－xSix合金在x = 0.25, 0.5, 0.75时的晶格参数及磁矩
Table 1. Lattice parameters and magnetic moments of Co2FeAl1－xSix alloys at x = 0.25, 0.5 and 0.75.
a/Å mAl/${{\text{μ}}_{\rm{B}}}$ mSi/${{\text{μ}}_{\rm{B}}}$ mFe/${{\text{μ}}_{\rm{B}}}$ mCo/${{\text{μ}}_{\rm{B}}}$ Mt/${{\text{μ}}_{\rm{B}}}$ Co2FeAl0.75Si0.25 5.6520 –0.053 –0.039 2.998 1.262 5.473 Co2FeAl0.5Si0.5 5.6607 –0.046 –0.028 3.037 1.337 5.688 Co2FeAl0.25Si0.75 5.6406 –0.038 –0.012 3.094 1.400 5.891 
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表 2 计算的Co2FeAl1－xSix (x = 0.25, x = 0.5, x = 0.75)合金的弹性常数、体模量及剪切模量
Table 2. The calculated cubic elastic constant C11, C12, C44, shear modulus Gv, GR and GH in GPa.
C11/GPa C12/GPa C44/GPa B/GPa GV/GPa GR/GPa GH/GPa Co2FeAl0.75Si0.25 247.38 166.97 142.33 193.77 101.48 70.60 86.04 Co2FeAl0.5Si0.5 266.15 143.57 141.73 184.43 109.55 92.94 101.25 Co2FeAl0.25Si0.75 176.46 51.042 137.67 92.85 107.69 93.14 100.42
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[1] Heusler F 1903 Deut. Phys. Ges. 5 219
[2] Murray S J, Marioni M, Allen S M, O’Handley R C 2000 Appl. Phys. Lett. 77 886
Google Scholar
[3] Donni A, Fischer P, Fauth F, Convert P, Aoki Y, Sugawara H, Sato H 1999 Physica B 259 705
[4] Wu G H, Yu C H, Meng L Q, Chen J L, Yang F M, Qi S R, Zhan W S 1999 Appl. Phys. Lett. 75 2990
Google Scholar
[5] Saha B, Shakouri A, Sands T D 2018 Appl. Phys. Rev. 5 021101
Google Scholar
[6] Webster P J 1971 J. Phys. Chem. Solids 32 1221
Google Scholar
[7] Kübler J, William A R, Sommers C B 1983 Phys. Rev. B 28 1745
Google Scholar
[8] de Groot R A, Müller F M, van Engen P G, Buschow K H J 1983 Phys. Rev. Lett. 50 2024
Google Scholar
[9] Comtesse D, Geisler B, Entel P, Kratzer P, Szunyogh L 2014 Phys. Rev. B 89 094410
Google Scholar
[10] Fecher G H, Felser C 2007 J. Phys. D: Appl. Phys. 40 1582
Google Scholar
[11] Li X M, Li T, Chen Z F, Hui F, Li X S, Wang X R, Xu J B, Zhu H W 2017 Appl. Phys. Rev. 4 021306
Google Scholar
[12] Balli M, Jandl S, Fournier P, Kedous-Lebouc A 2017 Appl. Phys. Rev. 4 021305
Google Scholar
[13] Kainuma R, Imano Y, Ito W, Sutou Y, Morito H, Okamoto S, Kitakami O, Oikawa K, Fujita A, Kanomata T, Ishida K 2006 Nature 439957
[14] Yu S Y, Liu Z H, Liu G D, Chen J L, Cao Z X, Wu G H, Zhang B, Zhang X X 2006 Appl. Phys. Lett. 89 162503
Google Scholar
[15] Dubenko I, Pathak A K, Stadler S, Ali N, Kovarskii Y, Prudnikov V N, Perov N S, Granovsky A B 2009 Phys. Rev. B 80 092408
Google Scholar
[16] Karaca H E, Karaman I, Basaran B, Ren Y, Chumlyakov Y I, Maier H J 2009 Adv. Funct. Mater. 19 983
Google Scholar
[17] Chmielus M, Zhang X X, Witherspoon C, Dunand D C, Mullner P 2009 Nat. Mater. 8 863
Google Scholar
[18] Sarawate N, Dapino M 2006 Appl. Phys. Lett. 88 121923
Google Scholar
[19] Mañosa L, González-Alonso D, Planes A, Bonnot E, Barrio M, Tamarit J L, Aksoy S, Acet M 2010 Nat. Mater. 9 478
Google Scholar
[20] Barman S R, Chakrabarti A, Singh S, Banik S, Bhardwaj S, Paulose P L, Chalke B A, Panda A K, Mitra A, Awasthi A M 2008 Phys. Rev. B 78 134406
Google Scholar
[21] Zayak A T, Entel P, Rabe K M, Adeagbo W A, Acet M 2005 Phys. Rev. B 72 054113
Google Scholar
[22] 罗礼进, 仲崇贵, 董正超, 方靖淮, 周朋霞, 江学范 2010 物理学报 59 8037
Google Scholar
Luo L J, Zhong C G, Dong Z C, Fang J H, Zhou P X, Jiang X F 2010 Acta Phys. Sin. 59 8037
Google Scholar
[23] 罗礼进, 仲崇贵, 江学范, 方靖淮, 蒋青 2010 物理学报 59 521
Google Scholar
Luo L J, Zhong C G, Jiang X F, Fang J H, Jiang Q 2010 Acta Phys. Sin. 59 521
Google Scholar
[24] 罗礼进, 仲崇贵, 赵永林, 方靖淮, 周朋霞, 江学范 2011 物理学报 60 127502
Google Scholar
Luo L J, Zhong C G, Zhao Y L, Fang J H, Zhou P X, Jiang X F 2011 Acta Phys. Sin. 60 127502
Google Scholar
[25] 罗礼进, 仲崇贵, 董正超, 方靖淮, 周朋霞, 江学范 2012 物理学报 61 207503
Google Scholar
Luo L J, Zhong C G, Dong Z C, Fang J H, Zhou P X, Jiang X F 2012 Acta Phys. Sin. 61 207503
Google Scholar
[26] Luo H Z, Jia P Z, Liu G D, Meng F B, Liu H Y, Liu E K, Wang W H, Wu G H, 2013 Solid State Commun. 17044
[27] Luo H Z, Meng F B, Liu G D, Liu H Y, Jia P Z, Liu E K, Wang W H, Wu G H 2013 Intermetallics 38 139
Google Scholar
[28] Kress G, Hafner J 1993 Phys. Rev. B 47 558
Google Scholar
[29] Perdew J P, Chevary J A, Vosko S H, Jackson K A, Pederson M R, Singh D J, Fiolhais C 1992 Phys. Rev. B 46 6671
Google Scholar
[30] Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
Google Scholar
[31] Kandpal H C, Fecher G H, Felser C 2007 J. Phys. D: Appl. Phys. 40 1507
Google Scholar
[32] Madsen G K H, Singh D J 2006 Comput. Phys. Commum. 175 67
Google Scholar
[33] Galanakis I, Mavropoulos P, Dederichs P H 2006 J. Phys. D: Appl. Phys. 39 765
Google Scholar
[34] Sargolzaei M, Richter M, Koepernik K, Opahle I, Eschrig H, Chaplygin I 2006 Phys. Rev. B 74 224410
Google Scholar
[35] Jansen H J F, Freeman A J 1984 Phys. Rev. B 30 561
Google Scholar
[36] Li J, Li J, Zhang Q, Zhang Z D, Yang G, Ma H R, Lu Z M, Fang W, Xie H X, Liang C Y, Yin F X 2016 Comp. Mater. Sci. 125 183
Google Scholar
[37] Li J, Yang G, Yang Y M, Ma H R, Zhang Q, Zhang Z D, Fang W, Yin F X, Li J 2017 J. Magn. Magn. Mater. 442 371
Google Scholar
[38] Kourov N I, Marchenkov V V, Perevozchikova Y A, Weber H W 2017 Phys. Solid State 59 898
Google Scholar
[39] Bilc D I, Mahanti S D, Kanatzidis M G 2006 Phys. Rev. B 74 125202
Google Scholar
[40] Al S, Arikan N, Demir S, Iyig€or A 2018 Physica B 531 16
Google Scholar
[41] Li J, Zhang Z D, Sun Y B, Zhang J, Zhou G X, Luo H Z, Liu G D 2013 Physica B 409 35
Google Scholar
[42] Okamura S, Miyazaki A, Sugimoto S 2005 Appl. Phys. Lett. 86 232503
Google Scholar
[43] Zhu W H, Wu D, Zhao B C, Zhu Z D, Yang X D, Zhang Z Z, Jin Q Y 2017 Phys. Rev. Appl. 8 034012
[44] Hazra B K, Raja M M, Srinath S 2016 J. Phys. D: Appl. Phys. 49 065007
Google Scholar
[45] Xu Z, Zhang Z, Hu F, Liu E, Xu F 2016 Mater. Res. Express 3 116103
Google Scholar
[46] Yadav A, Chaudhary S 2015 J. Appl. Phys. 118 193902
Google Scholar
[47] Chen J, Sakuraba Y, Masuda K, Miura Y, Li S, Kasai S, Furubayashi T, Hono K 2017 Appl. Phys. Lett. 110 242401
Google Scholar
[48] Huang X F, Dai Z W, Huang L, Lu G D, Liu M, Piao H G, Kim D H, Yu S C, Pan L Q 2016 J. Phys.: Condens. Matter 284 76006
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