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Review

The magnetism of Fe _ð1xÞ Co _x B alloys: First principle calculations ^$

P.H. Lee ^{a, } , Z.R. Xiao ^b , K.L. Chen ^c,d , Y. Chen ^e , S.W. Kao ^f , T.S. Chin ^f,g

aThe Affiliated Senior High School of National Taiwan Normal University, Taipei, Taiwan

bDepartment of Physics, National Taiwan University, Taipei, Taiwan

cTaipei Municipal Jianguo High School, Taipei, Taiwan

dThe MacDuffie School, Springfield, MA, USA

eDepartment of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

fDepartment of Material Science and Engineering, National Tsing Hua University, Taiwan

gDepartment of Materials Science and Engineering, Feng Chia University, Taichung, Taiwan

a r t i c l e i n f o

Article history:

Received 24 September 2008 Received in revised form 27 February 2009 Accepted 21 March 2009

PACS:

75.50.Bb 21.60.De 31.15.A

Keywords:

First-principle spin-polarized calculations Korringa-Kohn-Rostoker method Coherent potential approximation Virtual crystal approximation

a b s t r a c t

Calculations of magnetism are made on transition metal monoboride of Fe

ð1xÞ

Co

x

B, with the scale 0pxp1. In this paper, we calculate the ferromagnetic variation of transition metal monoboride by the method of virtual crystal approximation (VCA), based on density-functional theory (DFT) with generalized gradient approximation (GGA). The variations of ferromagnetism of Fe

ð1xÞ

Co

x

B alloys by the first-principle spin-polarized calculations are in agreement with experimental results, the prediction of the Stoner model and the Korringa–Kohn–Rostoker (KKR) method with coherent potential approxima- tion (CPA). The spontaneous magnetization decreases with increasing x and vanishes at around x ¼ 0:85.

This complies with earlier findings. The instability at around x ¼ 0:8 is elucidated by electron filling of the Fe 3d orbital both from valence and interstitial electrons.

Crown Copyright & 2009 Published by Elsevier B.V. All rights reserved.

Contents

1. Introduction . . . 1989

2. Method of calculation . . . 1990

3. Results and discussion. . . 1990

4. Conclusions . . . 1992

References . . . 1992

1. Introduction

The properties of transition metal alloys and transition metal borides have been under intense study for decades [1,2]. Among these properties, magnetization has been given special attention because of its application in the electronics industry. This

magnetism results from the partially filled valence d-orbital electrons, electronic structures, exchange correlations and various other factors. Previous studies have provided us with insights on the role of d-orbital electrons in iron borides, indicating that the electrons are hybridized in a subtle way [3]. In addition, the magnetic properties are shown to be correlated with the electronic structure [4], as well as the long-range spin interaction [5].

Various transition metal alloy and transition metal boride materials have been studied, in both experimental analyses and theoretical calculations, for example, Co

ð1xÞ

B

x

[6], Fe

ð1xÞ

B

x

[7,8], Fe

x

Mn

ð1xÞ

[9], Fe

n

B [10], Fe-based glass-forming alloys [11], FeCoB film [12,13] and Fe

x

Co

ð80xÞ

B

20

[14]. Li and Wang [15] calculated Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/physb

Physica B

0921-4526/$ - see front matter Crown Copyright & 2009 Published by Elsevier B.V. All rights reserved.

doi:10.1016/j.physb.2009.03.029

$Supported by the Ministry of Education, Taiwan, North K-12 of Nanotechnol- ogy Human Resource Development (NHRD) Program.

Corresponding author. Tel.: +886 2 27075215; fax: +886 2 27075133.

E-mail addresses:physics.lee@msa.hinet.net (P.H. Lee),d91222007@ntu.edu.tw (Z.R. Xiao),gmilk2008@gmail.com (K.L. Chen),chen.yi.first@gmail.com (Y. Chen), sw.kao@msa.hinet.net (S.W. Kao),tschin@fcu.edu.tw (T.S. Chin).

various properties of FeB and Fe

2

B with the method of linearized augmented plane wave (LAPW). Ching et al. [3] used the orthogonalized linear-combinations-of-atomic-orbitals method to investigate the magnetization of FeB, Fe

2

B and Fe

3

B. However, there is relatively little research done with the transition metal alloy borides Fe

x

Co

1x

B. Both FeB and CoB can have the same orthorhombic (Pnma) structure [16], and it is interesting to investigate the profile of the change in magnetic properties of the material if some of the Co atoms are replaced by Fe in CoB.

Klinduhov et al. have done pioneering investigations in the magnetic properties of the Fe

x

Co

1x

B using the Korringa–Kohn–

Rostoker (KKR) method with coherent potential approximation (CPA) [17,18]. The KKR method, also called Green’s function method, utilizes Green’s function in solving Schro¨dinger’s equa- tion and results in a nice formulation to calculate physical quantities [19]. CPA is a single-site approximation which pre- sumes that the average randomness in an alloy can be calculated by assuming a modified electron Green’s function which is the same throughout the crystal [20]. As shown by Klinduhov et al., this method yielded a good prediction in calculating the magnetism of Fe

x

Co

1x

B.

In contrast to the previous work by Klinduhov et al., in this paper, we have calculated the magnetic phase transition and the trend of magnetic moments versus cobalt concentration of the same alloy but with a different method, implemented by the wien2k package: full-potential linearized augmented plane wave method (FLAPW), based on the density-functional theory (DFT) with generalized gradient approximation (GGA) [21]. In order to approximate the effect of the Fe addition, virtual crystal approximation (VCA) is applied, which assumes that the averaged randomness in the alloy can be approximated with a virtual atom which has a non-integer atomic number. For example, in the case of Fe

0:5

Co

0:5

B alloys, the virtual transition metal atom will have atomic number 26.5.

In addition, we also relate the result of calculation to the Stoner model [22–25], which is used for explaining the mechanism of spontaneous spin polarization and the Slater–Pauling curve. The existence of a Slater–Pauling magnetization curve is well known in the transition metal compounds, specifically in the formula types T

2

B, TB and some phosphides, where T belongs to the first row of transition metal [26–28].

With the half-d-band occupied in compounds, such a curve describing saturation magnetization decreases linearly with a slope of 1 by progressively adding valence electrons to higher split half-d-band orbital. Because of the independence of the curve from crystal structure, we can draw a single Slater–Pauling curve to predict the magnetization for different crystal structures of transition metal borides. The question we want to analyze is then whether the magnetization of Fe

ð1xÞ

Co

x

B is dependent on a Slater–Pauling magnetization curve. In order to fully understand the characteristics of Fe and Co in Fe

ð1xÞ

Co

x

B, we treat the mixture of TM1 and TM2 as the singular TM in the boride compounds, TM as transition metal.

2. Method of calculation

In order to calculate electronic structures and magnetic properties of Fe

ð1xÞ

Co

x

B, we use the full-potential linearized augmented plane wave (FLAPW) method implemented by the wien2k packages [29]. It is beneficial to utilize ab initio methods to calculate various crystals and compounds [30]. For the method of virtual crystal approximation, the electron number we use to calculate the density of state (DOS) and magnetism is the mean electron number of the linear combination of TM1 and TM2. In this approximation, the nuclear and valence charges of the TM

atom are continuously altered from Fe ðz ¼ 26Þ to Co ðz ¼ 27Þ to represent different cases in Fe

ð1xÞ

Co

x

B. This combination of TM1 and TM2 will help reveal how ferromagnetism makes the transition to the state of non-ferromagnetism, meaning that it is an ideal case for us to recognize the role of the 3d electron in magnetization.

The crystal structures of FeB and CoB calculated here refer to Bjurstroem’s experimental work [16], while the space group is chosen to be Pnma instead of Pbnm. Under this symmetry, the lattice constants for FeB are a ¼ 5:495 ˚ A, b ¼ 2:946 ˚ A and c ¼ 4:053 ˚ A; and for CoB a ¼ 5:243 ˚ A, b ¼ 3:037 ˚ A and c ¼ 3:948 ˚ A. To reduce the complexity of electronic structures, the lattice constants of Fe

ð1xÞ

Co

x

B we used throughout the study are determined by the linear interpolation. For convenient comparison, all muffin–tin radii, RMT, for TM sites are set as 2.3 Bohr and the boron sites are set as 1.5 Bohr. In addition, the k- points grid is set to 7  13  10 for all calculations. The formula used for the calculation is Fe

ð1xÞ

Co

x

B and the RKM ðRMT  k

max

Þ value here is 7 in every case. The fraction x in the method of virtual crystal approximation plays the role of mean electron number. In TM1

ð1xÞ

TM2

x

B, Z ¼ Z

1

 ð1  xÞ þ Z

2

 x is the average of electron number, where Z

1

and Z

2

represent the electron number of TM1 and TM2, respectively.

The Stoner model we use here explains the dramatic change in magnetic behavior with the particular cobalt fraction in Fe

ð1xÞ

Co

x

B. It deals successfully with the prediction of magnetism of transition metal and implies the uniform susceptibility [25]

w ¼ N

TM

ðE

F

Þ 1  N

TM

ðE

F

Þ  I

TM

, (1)

where N

TM

ðE

F

Þ is the non-magnetic local density of states for each TM atom at Fermi level, and I

TM

¼ I

TM1

 ð1  xÞ þ I

TM2

 x is the average of Stoner constant, where I

TM1

and I

TM2

represent the Stoner constant of TM1 and TM2, respectively. If N

TM

ðE

F

Þ  I

TM

X1, the non-ferromagnetic state becomes unstable, which implies spontaneous magnetization.

3. Results and discussion

Fig. 1 shows the magnetic moment per TM atom in Fe

ð1xÞ

Co

x

B.

The moment of FeB is about 1:21 m

_B

=f:u: in agreement with the previous studies, while the calculated magnetic moments of Fe and B inside the muffin-tin radius are 1.31 and 0:04 m

B

, respectively. When the fraction of cobalt (x) becomes larger, the magnetic moment gradually decreases. It decreases to lower than 1:0 m

B

=f:u: when x is increased to 0.15. When x reaches 0.8, the magnetic moments for each TM, boride atom and formula unit are reduced to 0.147, 0:04 and 0:136 m

_B

, respectively. The relation between x and magnetic moments is almost linear with the slope of 1:35 m

_B

=f:u: This result agrees rather well with previous calculations done by Klinduhov et al. using the KKR–CPA method [17,18], as well as the experimental result obtained by Cadeville [32,33].

According to this trend, the moments become zero when x is larger than 0.9. However, our results from VCA calculations show that the moments come to zero when x is up to 0.85. The curve in the linear region of Fig. 1 is similar to that of Slater–Pauling curve, although the slope of the latter is 1 or 1. The larger magnitude of the slope in our data reflects an unusual mechanism appearing in Fe

ð1xÞ

Co

x

B. In order to determine why magnetic moment decreases with electron occupation, we plot total interstitial charge and majority (spin up) and minority (spin down) charges inside the TM atom in Fig. 2. As the fraction x increases from 0 to 0.8, the occupation of the minority state increases about 1.08 electrons/f.u., almost embedding inside a TM atom. The

P.H. Lee et al. / Physica B 404 (2009) 1989–1992

1990

changes of valence electrons provide the augmentation of 0.8 electrons in the minority state, and remarkably, the other increment of 0.28 electrons come partially from the interstitial region. Because, as the attraction of the TM nucleus becomes greater, the energy of the d band of TM atoms gets lower as x increases. This can explain why the electrons are transferred from the interstitial region to the TM minority band.

We also calculate the non-polarized DOS NðEÞ for all the systems of Fe

ð1xÞ

Co

x

B, and find the change in energy D EðmÞ after the formation of a ferromagnetic state. To compare the stability of magnetization for various cobalt fractions, we calculate the magnetization energy, the energy difference per formula unit between the ferromagnetic state and the non-ferromagnetic state.

The higher the magnetization energy, the more stable the ferromagnetic state is. Within the suitable criteria, the Stoner model and the FLAPW method indeed reveal the ferromagnetism for the mixture system of Fe

ð1xÞ

Co

x

B. Fig. 3 illustrates the trend in which the magnetization energy gradually decreases with the cobalt fraction x. When x ¼ 0, namely FeB, the magnetization energy is close to 165 meV/f.u. As seen in Fig. 3, there is a linear relationship with a slope of about 310 meV/f.u. when x is between 0 and 0.4, and as x value is larger than 0.4, the curve is more flat. If we extend the linear region across x axis, it would cross at 0.53.

Instead, the magnetization energy is close to zero as x reaches 0.8.

It is known that a ferromagnetic state would be reduced to non- ferromagnetic configuration when the magnetization energy reaches zero. As x40:85, the electronic structures of Fe

ð1xÞ

Co

x

B are identical for both ferromagnetic and non-ferromagnetic initial conditions for calculations. We observed different magnetic states, however, as x ¼ 0:8, even though the magnetization energy is almost zero. The results of VCA calculations indicate that it is possible for the two distinguishable magnetic states of Fe

ð1xÞ

Co

x

B to coexist near x ¼ 0:8.

According to the Stoner model, spontaneous magnetization appears when the condition N

TM

ðE

F

Þ  I

TM

41 is satisfied, where 0

Fraction of Co (x) 11.4

11.6 11.8 12 12.2 12.4 12.6 12.8 13 13.2 13.4

Electron occupation (states / f.u.)

TM spin-up TM spin-down

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2

interstitial

spin-up + spin-down

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 2. The calculated partial electron occupations per formula unit in Feð1xÞCoxB.

The total interstitial charge (right axis), majority (spin up) and minority (spin down) charges (left axis) inside TM atom are shown. The unusual increasing of minority electrons in TM is partly due to the contribution of interstitial electrons.

0

Fraction of Co (x) 0

20 40 60 80 100 120 140 160 180 200

Magnetization Energy (meV / f.u.)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 3. The magnetization energy per TM atom in Feð1xÞCoxB.

0

Fraction of Co (x) 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5

N

TM

(E

F

) (states / eV / f.u.)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

N

TM

(EF) × I

TM

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 4. The product of Stoner constant and total density of states (DOS). The dashed and solid lines represent NTMðEFÞ ITMand NTMðEFÞversus x, respectively.

Fig. 1. The circles represent the calculated magnetic moments per TM atom in Feð1xÞCoxB in this study. The dashed line is the Slater–Pauling curve with the slope of 1. The x value is defined as the fraction of Co atom in this system. The hollow square points are the experimental data taken from Lemius and Kuentzler [31]. The triangle points are the result of the KKR–CPA calculation carried out by Klinduhov et al. [17,18]. As shown in the figure, the results calculated with virtual crystal approximation agree pretty well with experimental result and KKR–CPA calcula- tions.

N

TM

ðE

F

Þ is the non-ferromagnetic partial DOS of TM atom at Fermi level, and I

TM

is the Stoner exchange parameter. For different x in the Fe

ð1xÞ

Co

x

B, the Stoner parameter of the TM atom is determined by linear interpolation of each atom, shown by the relation of N

TM

ðE

F

Þ  I

TM

values versus x in Fig. 4. The I

TM

for Fe and Co are 0.46 and 0.47 eV, respectively [25]. Based on the prediction of the Stoner model, FeB illustrates the spontaneously spin-polarized state because the N

TM

ðE

F

Þ  I

TM

is 1.84, larger than the necessary value 1. The N

TM

ðE

F

Þ  I

TM

value decreases gradually with increasing cobalt fraction x. When x is near 0.75, the curve will cross the line N

TM

ðE

F

Þ  I

TM

¼ 1. Under the condition of xo0:75, the Fe

ð1xÞ

Co

x

B is ferromagnetic, and it is one kind of non- ferromagnetic if x is greater than 0.75. Remarkably, the N

TM

ðE

F

Þ  I

TM

value is slightly less than 1 when x ¼ 0:8, even though the ferromagnetic state exists. Consistently, the energy difference between magnetic and non-ferromagnetic states is almost zero for x ¼ 0:8. This suggests that spontaneous magnetization may not occur for Fe

0:2

Co

0:8

B, and the ferromagnetic state might be metastable. Such instability is attributed to the fact that electron filling of TM minority state, which causes the decrease in magnetic moments, comes not only from the changes in valence electrons, but also from the attracted interstitial electrons.

By considering the experimental data and the FLAPW calcula- tion illustrated in Fig. 1, we find from both sources that FeB (x ¼ 0) has a higher magnetic moment than that of CoB (x ¼ 1). The experimental data of the magnetic moment for FeB and CoB are 1.12 and 0 m

_B

=f:u:, respectively. The detailed results are listed in Table 1, in which the moment calculation from FLAPW matches very well with the experimental data. Rather than undertake the complex analysis by FLAPW calculation, the Stoner model serves satisfactorily in the prediction of ferromagnetism within certain criteria.

4. Conclusions

In this paper, we discussed magnetic properties of Fe

ð1xÞ

Co

x

B using virtual crystal approximation. The magnetic moments, magnetization energy and density of states of Fe

ð1xÞ

Co

x

B are calculated based on spin-polarized density functional theory with generalized gradient approximation implemented by the wien2k package. As the cobalt fraction x increases, magnetic moment decreases almost linearly. It drops to zero as x is larger than 0.85.

When x is near 0.8, the transition from a ferromagnetic to a non- ferromagnetic state occurs. The electrons filling of the TM minority state causes the decrease of magnetic moments of Fe

ð1xÞ

Co

x

B with increasing x. The source of filling electrons comes not only from the changes of valence electrons, but also from the attracted interstitial electrons. We also found that the Stoner model can predict that magnetic transition for Fe

ð1xÞ

Co

x

B occurs near x ¼ 0:8.

References

[1] P. Mohn, D.G. Pettifor, J. Phys. C Solid State Phys. 21 (1988) 2829.

[2] A.P. Malozemoff, A.R. Williams, V.L. Moruzzi, Phys. Rev. B 29 (1984) 1620.

[3] W.Y. Ching, Y.-N. Xu, B.N. Harmon, J. Ye, T.C. Leung, Phys. Rev. B 42 (1990) 4460.

[4] Ch. Hausleitner, J. Hafner, Phys. Rev. B 47 (1993) 5689.

[5] W. Zhong, G. Overney, D. Toma´nek, Phys. Rev. B 47 (1993) 95.

[6] H. Tanaka, S. Takayama, M. Hasegawa, T. Fukunaga, U. Mizutani, A. Fujita, K. Fukamichi, Phys. Rev. B 47 (1993) 2671.

[7] J. Hafner, M. Tegze, Ch. Becker, Phys. Rev. B 49 (1994) 285.

[8] J.W. Taylor, J.A. Duffy, A.M. Bebb, M.J. Cooper, S.B. Dugdale, J.E. McCarthy, D.N.

Timms, D. Greig, Y.B. Xu, Phys. Rev. B 63 (2001) 220404.

[9] C. Jing, S.X. Cao, J.C. Zhang, Phys. Rev. B 68 (2003) 224407.

[10] A.M. Bratkovsky, S.N. Rashkeev, G. Wendin, Phys. Rev. B 48 (1993) 6260.

[11] M. Mihalkovic´, M. Widom, Phys. Rev. B 70 (2004) 144107.

[12] C.L. Platt, M.K. Minor, T.J. Klemmer, IEEE Trans. Magnetics 37 (4) (2001) 2302.

[13] M. Kevin Minor, T.M. Crawford, T.J. Klemmer, Y. Peng, D.E. Laughlin, J. Appl.

Phys. 91 (10) (2002) 8453.

[14] W. Kettler, R. Wernhardt, M. Rosenberg, J. Appl. Phys. 53 (11) (1982) 8248.

[15] G. Li, D. Wang, J. Phys: Condens. Matter 1 (1989) 1799.

[16] T. Bjurstroem, A. Kemi, Mineral Geol. A 11 (1933) 1.

[17] N. Klinduhov, M.G. Shelyapina, V.S. Kasperovich, E.K. Hlil, D. Fruchart, J. Magn.

Magn. Mater. 300 (2006) 563.

[18] N.A. Klindukhov, V.S. Kasperovich, M.G. Shelyapina, H.El. Kebir, Phys. Solid State 50 (2008) 302.

[19] W. Kohn, N. Rostoker, Phys. Rev. 94 (1954) 1111.

[20] P. Soven, Phys. Rev. 156 (1967) 809.

[21] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865.

[22] E.C. Stoner, Proc. Roy. Soc. London Ser. A 169 (1939) 339.

[23] P.M. Marcus, V.L. Moruzzi, Phys. Rev. B 38 (1988) 6949.

[24] G. Stollhoff, A.M. Oles´, V. Heine, Phys. Rev. B 41 (1990) 7028.

[25] J.F. Janak, Phys. Rev. B 16 (1977) 255.

[26] J.C. Slater, J. Appl. Phys. 8 (1937) 385.

[27] Y.S. Tyan, L.E. Toth, J. Elec. Mater. 3 (1) (1974) 1.

[28] L. Pauling, Phys. Rev. 54 (1938) 899.

[29] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, Computer code WIEN2K, Vienna University of Technology, Austria, 2001.

[30] M. Fuchs, M. Scheffler, Comput. Phys. Comm. 119 (1999) 67.

[31] B. Lemius, R. Kuentzler, J. Phys. F: Met. Phys. 10 (1980) 155.

[32] M.C. Cadeville, Ph.D. Thesis, Strasbourg University, 1965.

[33] M.C. Cadeville, I. Vincze, J. Phys. F: Met. Phys. 5 (1975) 790.

Table 1

The experimental and calculated data by FLAPW and Stoner criterion.

M (exp.) (

m

_B/f.u.) M (FLAPW) (

m

_B/f.u.) ITM[25](eV) NTMðEFÞ ITM(electrons/f.u.)

FeB 1:12 1:20 0:46 1:87

CoB 0:00 0:00 0:47 0:18

P.H. Lee et al. / Physica B 404 (2009) 1989–1992 1992