Heusler合金Co_2FeSi（001）表面的电子结构，磁性和半金属特性：第一性原理研究
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摘要
自旋电子学是近年来发展起来的一门新型交叉学科,主要利用电子的自旋来传输和存储信息.在自旋电子学领域,半金属铁磁体因为能够极大的提高从铁磁体注入电子到半导体时的自旋极化率,因而受到了科学家的广泛关注.Full-Heusler合金Co2FeSi在目前为止所报道的众多半金属铁磁体材料中具有最高的磁矩和居里温度,而被认为是理想的自旋电子注入源之一
本文利用全势线性缀加平面波方法(full-potential linearized augmented plane wave mothod, FLAPW),结合广义梯度近似(generalized gradient approximation,GGA)并考虑晶格点有效的库仑交换作用U,研究了Heusler合金Co2FeSi (001)表面的电子结构,磁性以及半金属特性.沿[001]方向考虑了两种可能的表面,即Co原子终止的表面与Fe和Si原子终止的表面.
计算结果表明：
(1)Co原子终止的表面将失去半金属特性.这是由于表面的产生导致表面层Co原子的配位数减少,其自旋向下的态密度发生局域化,并从低能级向高能级移动,破坏表面层Co原子自旋向下子带的带隙.表面层自旋极化率分别为-48.3%(U7.5)和-43.2%(U10).表面层Co原子的自旋磁矩(～1.75μB)与中间上一层Co原子的自旋磁矩(～1.51μB)相比有了明显增加,是因为表面层Co原子的配位数减少,使表面层Co原子的交换劈裂有所增加.表面下一层Fe原子的自旋磁矩(～3μB,)与中间层Fe原子的自旋磁矩(～3.15μB,)相比有所减小,这是由于表面下一层Fe原子与表面层Co原子杂化增强所致.
(2)Fe和Si原子终止的表面也失去了半金属特性.这是由于表面的产生使表面层Fe原子的最近邻Co原子的数目由原来的八个减少至四个,因此未填充的自旋向下的态由高能级向低能级移动,同时能隙出现了表面态.表面层自旋极化率分别为-87.3%(U7.5)和-43.2%(U10).表面层Fe原子的自旋磁矩(～3.28μB)大于中间层自旋磁矩(～3.14μB),是因为表面层Fe原子的配位数减少使表面层Fe原子的交换劈裂有所增加.表面下一层和中间上一层的Co原子的自旋磁矩基本上没有变化.
Spintronics is a new interactive-type subject in recent years; it mainly explores the transport and storage of information by electron spin. In spintronics, the Half-metallic alloy attracts great attentions, because it could improve electron spin polarization when the alloy injects into the semiconductor. In present, the Full-Heusler alloy Co2FeSi has the highest magnetic moment and Curie temperature in the reported many ferromagnet materials. It is considered an ideal source of spin injection.
In this paper, by the full-potential linearized augmented plane wave (FLAPW) method within the generalized gradient approximation (GGA) and plus on-site coulo-mb exchange interaction U, we have investigated the electronic structures, magnetism and half-metallicity of the Full-Heusler alloy Co2FeSi (001) surface. For [001] directi-on, we have considered two types of surfaces:the Co atoms terminated and the Fe and Si atoms terminated surfaces. The results are followings:
(1) Co atoms terminate surface will loss the half-metallicity. It was found that the minority-spin states of Co (surface) atoms were localized and moved to higher energy from lower energy, which destroys the minority spin band gap due to the reduced coordination number of Co (surface) atoms. The calculated spin polarizations of the surface layer are negative with values of-48.3%(U75) and-43.2%(U10). The magnetic moment (～1.75μB) of the surface Co atom is larger than that (～1.51μB) of the inner layer because of the reduced coordination number of Co (surface) atom, which increases the exchange splitting at the surface. The magnetic moment (～3μB) of the subsurface Fe atom decreased slightly than that (～3.15μB) of the center layer due to increased covalent hybridization between the Fe (subsurface) and the Co (surface) atoms.
(2) Fe and Si atoms terminated surface also loss the half-metallicity. At the surface the coordination number of Fe (surface) atoms was reduced from 8 to 4 Co atoms. It was found that the unoccupied minority-spin states of surface Fe atoms shifted from higher energy to lower energy, and the surface states appear in the band gap. The calculated spin polarizations for the surface layers are negative with values of-87.3%(U7.5) and-43.2%(U10). The magnetic moment (～3.28μB) of surface Fe is slightly larger than the value (～3.14μB) of the inner layer due to the increase of the exchange splitting at the surface. On the other hand, the magnetic moment of subsurface Co atom is similar to that of the inner layer Co atom.
本文利用全势线性缀加平面波方法(full-potential linearized augmented plane wave mothod, FLAPW),结合广义梯度近似(generalized gradient approximation,GGA)并考虑晶格点有效的库仑交换作用U,研究了Heusler合金Co2FeSi (001)表面的电子结构,磁性以及半金属特性.沿[001]方向考虑了两种可能的表面,即Co原子终止的表面与Fe和Si原子终止的表面.
计算结果表明：
(1)Co原子终止的表面将失去半金属特性.这是由于表面的产生导致表面层Co原子的配位数减少,其自旋向下的态密度发生局域化,并从低能级向高能级移动,破坏表面层Co原子自旋向下子带的带隙.表面层自旋极化率分别为-48.3%(U7.5)和-43.2%(U10).表面层Co原子的自旋磁矩(～1.75μB)与中间上一层Co原子的自旋磁矩(～1.51μB)相比有了明显增加,是因为表面层Co原子的配位数减少,使表面层Co原子的交换劈裂有所增加.表面下一层Fe原子的自旋磁矩(～3μB,)与中间层Fe原子的自旋磁矩(～3.15μB,)相比有所减小,这是由于表面下一层Fe原子与表面层Co原子杂化增强所致.
(2)Fe和Si原子终止的表面也失去了半金属特性.这是由于表面的产生使表面层Fe原子的最近邻Co原子的数目由原来的八个减少至四个,因此未填充的自旋向下的态由高能级向低能级移动,同时能隙出现了表面态.表面层自旋极化率分别为-87.3%(U7.5)和-43.2%(U10).表面层Fe原子的自旋磁矩(～3.28μB)大于中间层自旋磁矩(～3.14μB),是因为表面层Fe原子的配位数减少使表面层Fe原子的交换劈裂有所增加.表面下一层和中间上一层的Co原子的自旋磁矩基本上没有变化.
Spintronics is a new interactive-type subject in recent years; it mainly explores the transport and storage of information by electron spin. In spintronics, the Half-metallic alloy attracts great attentions, because it could improve electron spin polarization when the alloy injects into the semiconductor. In present, the Full-Heusler alloy Co2FeSi has the highest magnetic moment and Curie temperature in the reported many ferromagnet materials. It is considered an ideal source of spin injection.
In this paper, by the full-potential linearized augmented plane wave (FLAPW) method within the generalized gradient approximation (GGA) and plus on-site coulo-mb exchange interaction U, we have investigated the electronic structures, magnetism and half-metallicity of the Full-Heusler alloy Co2FeSi (001) surface. For [001] directi-on, we have considered two types of surfaces:the Co atoms terminated and the Fe and Si atoms terminated surfaces. The results are followings:
(1) Co atoms terminate surface will loss the half-metallicity. It was found that the minority-spin states of Co (surface) atoms were localized and moved to higher energy from lower energy, which destroys the minority spin band gap due to the reduced coordination number of Co (surface) atoms. The calculated spin polarizations of the surface layer are negative with values of-48.3%(U75) and-43.2%(U10). The magnetic moment (～1.75μB) of the surface Co atom is larger than that (～1.51μB) of the inner layer because of the reduced coordination number of Co (surface) atom, which increases the exchange splitting at the surface. The magnetic moment (～3μB) of the subsurface Fe atom decreased slightly than that (～3.15μB) of the center layer due to increased covalent hybridization between the Fe (subsurface) and the Co (surface) atoms.
(2) Fe and Si atoms terminated surface also loss the half-metallicity. At the surface the coordination number of Fe (surface) atoms was reduced from 8 to 4 Co atoms. It was found that the unoccupied minority-spin states of surface Fe atoms shifted from higher energy to lower energy, and the surface states appear in the band gap. The calculated spin polarizations for the surface layers are negative with values of-87.3%(U7.5) and-43.2%(U10). The magnetic moment (～3.28μB) of surface Fe is slightly larger than the value (～3.14μB) of the inner layer due to the increase of the exchange splitting at the surface. On the other hand, the magnetic moment of subsurface Co atom is similar to that of the inner layer Co atom.
引文
[1]黄昆.固体物理学.北京,高等教育出版社,1998
[2]Prinz G A. Spin polarized transport. Phys. Today,1995,48:58(1-6)
[3]Ohno H, Munekata H, Penney T, et al. Magnetotransport properties of p-type(In, Mn) as diluted magnetic III-V semiconductors. Phys. Rev. Lett,1991,68: 2664(1-4)
[4]Prinz G A. Magnetoelectronics. Science,1998,282:1660(1-4)
[5]Pickett W E, Moodera J S. Half-matellic magnets. Phys. Today,2001,54:39(1-6)
[6]Baibich M N. Giant Magnetoresistance of (001) Fe/(001) Cr magnetic superlattices. Phys. Rev. Lett,1988,61:2472(1-4)
[7]Binasch G, Grunberg P, Saurenbach F, et al. Enhanced magnetoresistanc in layered magnetic structures with antiferromagnetic interlayer exchange. Phys. Rev. B,1989,39:4828(1-3)
[8]Moodera J S, Kinder L R, Wong T M, et al. Large magnetoresistance at room temperature in ferromagnetic thin film tunnel junctions. Phys. Rev. Lett,1995, 74:3273(1-4)
[9]Maekawa S G, fvert U. Electron tunneling between ferromagnetic films. IEEE Transactions on Magnetics,1982,3:707(1-2)
[10]Miyazaki T, Tezuka N. Giant magnetic tunneling effect in Fe/Al2O3/Fe junction. J. Magn. Magn. Mater,1995,139:231(1-4)
[11]Soulen Jr R J, Byers J M, Osofsky M S, et al. Measuring the spin polarization of a metal with a superconducting point contact. Science,1998,282:85(1-4)
[12]Monsma D J, Parkin S P. Spin polarization of tunneling current from ferromagnet/Al2O3 interfaces using copper-doped aluminum superconducting flms. Appl. Phys. Lett,2000,77:720(1-3)
[13]Groot R A, Mueller F M, Van Engen P G, et al. New class of materials: Half-metallic ferromagnets. Phys. Rev. Lett,1983,50:2024(1-4)
[14]Watts S M, Wirth S, Molnar S, el al. Evidence for two-band magnetotransport in Half-metallic chromium dioxide. Phys. Rev. B,2000,61:163(1-8)
[15]Park J H, Vescovo E, Kim H J, el al. Direct evidence for a Half-metallic ferroma-gnet, Nature,1998,392:794(1-3)
[16]Coey J M D, Viret M, Molnar S. Mixed-valence manganites. Adv. Phys,1999,48: 167(1-127)
[17]Steven P L, Philip B A, Taizo S. Band structure and transport properties of CrO2. Phys. Rev. B,1997,55:10253(1-8)
[18]Kamper K P, Schmitt W, Guntherodt G. CrO2-a new Half-metallic ferromagnet. Phys. Rev. Lett,1987,59:2788(1-4)
[19]Anderson P W. Electromagnetic Theory of Cyclotron Resonance in Metals. Phys. Rev,1955,100:749(1-2)
[20]Lyanda Geller Y. Magnetotransport in double perovskite Sr2FeMnO6:role of magnetic and nonmagnetic disorder. Phys. Rev. B,2001,64:214407 (1-4)
[21]Dedkov Yu S, Rudiger U, Guntherodt G. Evidence for the Half-metallic ferro-magnetic state of Fe3O4 by spin-resolved photoelectron spectroscopy. Phys. Rev. B,2002,65:64417(1-5)
[22]Abid M, Abid J P, Jannin S. Magnetotransport properties depending on the nan-ostructure of Fe3O4 nanowires. J. Phys.Condens. Matter,2006,8:6085(1-9)
[23]Galanakis I, Mavropoulos P. Zinc-blende compounds of transition elements with N, P, As, Sb, S, Se, and Te as Half-metallic systems. Phys. Rev. B,2003,67: 104417(1-8)
[24]Galanakis I. Surface Half-metallicity of CrAs in the zinc-blende structure. Phys. Rev. B,2002,66:12406(1-4)
[25]Picozzi S, Continenza A, Freeman A J. Co2Mn X (X=Si, Ge, Sn) Heusler compounds:ab initio study of their structural, electronic, and magnetic properties at zero and elevated pressure. Physical Review B,2002,66: 94421(1-9)
[26]Correa J S, Ch. Rangelov Eibl G, Braun J, et al. Surface electronic structure of NiMnSb(001). Phys. Rev. B,2006,73:125316(1-6)
[27]Wurmehl S, Fecher G H, Kandpal H C, et al. Geometric, electronic, and magnet-ic structure of Co2FeSi:Curie temperature and magnetic moment measurements and calculations. Phys. Rev. B,2005,72:184434(1-9)
[28]Ebrahim Hazrati, Hashemifar S Javad, Akbarzadeh Hadi. First principles study of bulk CrSe and CrSe/ZnSe(001) interface. Journal of Applied Physics,2008,104: 113719(1-6)
[29]Kobayashi, Cabral N J, Sousa J R. Critical properties of thin quantum and classical heisenbergms. Phys. Rev. B,2002,66:64417(1-6)
[30]吴玉蓉,胡望宇,赵栋梁.Heusler合金Ni2MnGa磁性微观机理的第一性原理.第十届全国青年材料科学技术研讨会.长沙,2005
[31]张炜,千正男,隋郁.Heusler合金Co2TiSn的磁性与输运性能.物理学报, 2005,10:4879(1-5)
[32]Dedcov Y S, Rudiger U, Guntherodt G. Evidence for the half-metallic ferromagnetic state of Fe3O4 by spin-resolved photoelectron spectroscopy. Phys. Rev. B,2002,65:64417(1-5)
[33]Inomata K. Structural dependence of the tunnel magnetoresistance for magnetic tunnel junctions with a full-Heusler Co2Fe(Al, Si) electrode. J. Phys. D, Appl. Phys,2006,39:816(1-8)
[34]Zarei S, Hashemifar S J, Akbarzadeh H. Half-metallicity at the Heusler alloy Co2Cr0.5Fe0.5Al(001) surface and its interface with GaAs(001). Journal of Physics. Condensed Matter,2009,21:55002(1-7)
[35]Hashemifar S J, Peter Kratzer, Matthias S. Preserving the Half-metallicity at the Heusler alloy Co2MnSi(001) surface:a density functional theory study. Phys. Rev. Lett,2005,94:96402(1-4)
[36]Wurmehl S, Fecher G H, Kandpal H C, et al. Investigation of Co2FeSi:The Heusler compound with highest curie temperature and magnetic moment. Appl. Phys. Lett,2006,88:32503(1-3)
[37]Hashimoto M, Herfort J, Trampert A, et al. Atomic ordering and interlayer diffusion of Co2FeSi films grown on GaAs(001) studied by transmission electron microscopy. J Vac. Sci. Technol,2007,25:1453(1-7)
[38]Gercsi Z, Rajanikanth A, Takahashi Y K, et al. Spin polarization of Co2FeSi full-Heusler alloy and tunneling magnetoresistance of its magnetic tunneling junctions. Applied Physics Letters,2006,8:82512(1-3)
[39]Kandpal H, Fecher G H, Felser C, et al. Correlation in the transition-metal-based Heusler compounds Co2MnSi and Co2FeSi. Phys.Rev. B,2006,73:94422 (1-11)
[40]Herman F, Dyke J P, Ortenberger I P. Improved statistical exchang approximation for inhomogeneous many-electron systems. Phys. Rev. Lett,1969,22:807(1-5)
[41]谢希德,陆栋主编.固体能带理论.上海,复旦大学出版社,1998
[42]李正中,固体理论.北京,高等教育出版社,1985
[43]Wimmer E, Krakauer H, Weinert M, et al. Full-potential self-consistent linearized-augmented-plane-wave method for calculating the electronic structure of molecules and surfaces:O2 molecule. Phys. Rev. B,1981,24: 864(1-12)
[44]Weinert M, Wimmer E, Freeman A J. Total-energy all-electron density functional method for bulk solids and surfaces. Phys. Rev. B,1982,26:4571(1-8)
[45]Perdew J P, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Phys. Rev. Lett,1996,77:3865(1-4)
[46]Svendsen P S, Barth U. Gradient expansion of the exchange energy from second-order density response theory. Phys. Rev. B,1996,54:17402(1-12)
[47]http://www. flapw. de
[48]Hohenberg P, Kohn W. Inhomogeneous electron gas. Phys. Rev. B,1964,136: 864(1-8)
[49]Kohn W, Sham L J. Self-consistent equation including exchange and correlation effects. Phys. Rev. A,1965,140:1133(1-6)
[50]Kohn W. Nobel Lecture:Electronic structure of matter-wave functions and density of functionals. Rev. Mod. phys,1998,71:1253(1-14)
[51]Slater J C. Wave Functions in a Periodic Potential. Phys. Rev, B,1937,51: 846(1-6)
[2]Prinz G A. Spin polarized transport. Phys. Today,1995,48:58(1-6)
[3]Ohno H, Munekata H, Penney T, et al. Magnetotransport properties of p-type(In, Mn) as diluted magnetic III-V semiconductors. Phys. Rev. Lett,1991,68: 2664(1-4)
[4]Prinz G A. Magnetoelectronics. Science,1998,282:1660(1-4)
[5]Pickett W E, Moodera J S. Half-matellic magnets. Phys. Today,2001,54:39(1-6)
[6]Baibich M N. Giant Magnetoresistance of (001) Fe/(001) Cr magnetic superlattices. Phys. Rev. Lett,1988,61:2472(1-4)
[7]Binasch G, Grunberg P, Saurenbach F, et al. Enhanced magnetoresistanc in layered magnetic structures with antiferromagnetic interlayer exchange. Phys. Rev. B,1989,39:4828(1-3)
[8]Moodera J S, Kinder L R, Wong T M, et al. Large magnetoresistance at room temperature in ferromagnetic thin film tunnel junctions. Phys. Rev. Lett,1995, 74:3273(1-4)
[9]Maekawa S G, fvert U. Electron tunneling between ferromagnetic films. IEEE Transactions on Magnetics,1982,3:707(1-2)
[10]Miyazaki T, Tezuka N. Giant magnetic tunneling effect in Fe/Al2O3/Fe junction. J. Magn. Magn. Mater,1995,139:231(1-4)
[11]Soulen Jr R J, Byers J M, Osofsky M S, et al. Measuring the spin polarization of a metal with a superconducting point contact. Science,1998,282:85(1-4)
[12]Monsma D J, Parkin S P. Spin polarization of tunneling current from ferromagnet/Al2O3 interfaces using copper-doped aluminum superconducting flms. Appl. Phys. Lett,2000,77:720(1-3)
[13]Groot R A, Mueller F M, Van Engen P G, et al. New class of materials: Half-metallic ferromagnets. Phys. Rev. Lett,1983,50:2024(1-4)
[14]Watts S M, Wirth S, Molnar S, el al. Evidence for two-band magnetotransport in Half-metallic chromium dioxide. Phys. Rev. B,2000,61:163(1-8)
[15]Park J H, Vescovo E, Kim H J, el al. Direct evidence for a Half-metallic ferroma-gnet, Nature,1998,392:794(1-3)
[16]Coey J M D, Viret M, Molnar S. Mixed-valence manganites. Adv. Phys,1999,48: 167(1-127)
[17]Steven P L, Philip B A, Taizo S. Band structure and transport properties of CrO2. Phys. Rev. B,1997,55:10253(1-8)
[18]Kamper K P, Schmitt W, Guntherodt G. CrO2-a new Half-metallic ferromagnet. Phys. Rev. Lett,1987,59:2788(1-4)
[19]Anderson P W. Electromagnetic Theory of Cyclotron Resonance in Metals. Phys. Rev,1955,100:749(1-2)
[20]Lyanda Geller Y. Magnetotransport in double perovskite Sr2FeMnO6:role of magnetic and nonmagnetic disorder. Phys. Rev. B,2001,64:214407 (1-4)
[21]Dedkov Yu S, Rudiger U, Guntherodt G. Evidence for the Half-metallic ferro-magnetic state of Fe3O4 by spin-resolved photoelectron spectroscopy. Phys. Rev. B,2002,65:64417(1-5)
[22]Abid M, Abid J P, Jannin S. Magnetotransport properties depending on the nan-ostructure of Fe3O4 nanowires. J. Phys.Condens. Matter,2006,8:6085(1-9)
[23]Galanakis I, Mavropoulos P. Zinc-blende compounds of transition elements with N, P, As, Sb, S, Se, and Te as Half-metallic systems. Phys. Rev. B,2003,67: 104417(1-8)
[24]Galanakis I. Surface Half-metallicity of CrAs in the zinc-blende structure. Phys. Rev. B,2002,66:12406(1-4)
[25]Picozzi S, Continenza A, Freeman A J. Co2Mn X (X=Si, Ge, Sn) Heusler compounds:ab initio study of their structural, electronic, and magnetic properties at zero and elevated pressure. Physical Review B,2002,66: 94421(1-9)
[26]Correa J S, Ch. Rangelov Eibl G, Braun J, et al. Surface electronic structure of NiMnSb(001). Phys. Rev. B,2006,73:125316(1-6)
[27]Wurmehl S, Fecher G H, Kandpal H C, et al. Geometric, electronic, and magnet-ic structure of Co2FeSi:Curie temperature and magnetic moment measurements and calculations. Phys. Rev. B,2005,72:184434(1-9)
[28]Ebrahim Hazrati, Hashemifar S Javad, Akbarzadeh Hadi. First principles study of bulk CrSe and CrSe/ZnSe(001) interface. Journal of Applied Physics,2008,104: 113719(1-6)
[29]Kobayashi, Cabral N J, Sousa J R. Critical properties of thin quantum and classical heisenbergms. Phys. Rev. B,2002,66:64417(1-6)
[30]吴玉蓉,胡望宇,赵栋梁.Heusler合金Ni2MnGa磁性微观机理的第一性原理.第十届全国青年材料科学技术研讨会.长沙,2005
[31]张炜,千正男,隋郁.Heusler合金Co2TiSn的磁性与输运性能.物理学报, 2005,10:4879(1-5)
[32]Dedcov Y S, Rudiger U, Guntherodt G. Evidence for the half-metallic ferromagnetic state of Fe3O4 by spin-resolved photoelectron spectroscopy. Phys. Rev. B,2002,65:64417(1-5)
[33]Inomata K. Structural dependence of the tunnel magnetoresistance for magnetic tunnel junctions with a full-Heusler Co2Fe(Al, Si) electrode. J. Phys. D, Appl. Phys,2006,39:816(1-8)
[34]Zarei S, Hashemifar S J, Akbarzadeh H. Half-metallicity at the Heusler alloy Co2Cr0.5Fe0.5Al(001) surface and its interface with GaAs(001). Journal of Physics. Condensed Matter,2009,21:55002(1-7)
[35]Hashemifar S J, Peter Kratzer, Matthias S. Preserving the Half-metallicity at the Heusler alloy Co2MnSi(001) surface:a density functional theory study. Phys. Rev. Lett,2005,94:96402(1-4)
[36]Wurmehl S, Fecher G H, Kandpal H C, et al. Investigation of Co2FeSi:The Heusler compound with highest curie temperature and magnetic moment. Appl. Phys. Lett,2006,88:32503(1-3)
[37]Hashimoto M, Herfort J, Trampert A, et al. Atomic ordering and interlayer diffusion of Co2FeSi films grown on GaAs(001) studied by transmission electron microscopy. J Vac. Sci. Technol,2007,25:1453(1-7)
[38]Gercsi Z, Rajanikanth A, Takahashi Y K, et al. Spin polarization of Co2FeSi full-Heusler alloy and tunneling magnetoresistance of its magnetic tunneling junctions. Applied Physics Letters,2006,8:82512(1-3)
[39]Kandpal H, Fecher G H, Felser C, et al. Correlation in the transition-metal-based Heusler compounds Co2MnSi and Co2FeSi. Phys.Rev. B,2006,73:94422 (1-11)
[40]Herman F, Dyke J P, Ortenberger I P. Improved statistical exchang approximation for inhomogeneous many-electron systems. Phys. Rev. Lett,1969,22:807(1-5)
[41]谢希德,陆栋主编.固体能带理论.上海,复旦大学出版社,1998
[42]李正中,固体理论.北京,高等教育出版社,1985
[43]Wimmer E, Krakauer H, Weinert M, et al. Full-potential self-consistent linearized-augmented-plane-wave method for calculating the electronic structure of molecules and surfaces:O2 molecule. Phys. Rev. B,1981,24: 864(1-12)
[44]Weinert M, Wimmer E, Freeman A J. Total-energy all-electron density functional method for bulk solids and surfaces. Phys. Rev. B,1982,26:4571(1-8)
[45]Perdew J P, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Phys. Rev. Lett,1996,77:3865(1-4)
[46]Svendsen P S, Barth U. Gradient expansion of the exchange energy from second-order density response theory. Phys. Rev. B,1996,54:17402(1-12)
[47]http://www. flapw. de
[48]Hohenberg P, Kohn W. Inhomogeneous electron gas. Phys. Rev. B,1964,136: 864(1-8)
[49]Kohn W, Sham L J. Self-consistent equation including exchange and correlation effects. Phys. Rev. A,1965,140:1133(1-6)
[50]Kohn W. Nobel Lecture:Electronic structure of matter-wave functions and density of functionals. Rev. Mod. phys,1998,71:1253(1-14)
[51]Slater J C. Wave Functions in a Periodic Potential. Phys. Rev, B,1937,51: 846(1-6)