Theoretical study of diluted Co/Cu alloys

Journal of Magnetism and Magnetic Materials 305 (2006) 428–431

Theoretical study of diluted Co/Cu alloys

Shih-Jye Sun



, Shin-Pon Ju

, Yu-Chieh Lo

a_{Department of Applied Physics, National University of Kaohsiung, Kaohsiung 811, Taiwan}

b_{Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan}

Received 21 October 2005; received in revised form 14 January 2006 Available online 6 March 2006

Abstract

This study employs the self-consistent Green’s function method to study the magnetic properties of diluted CoxCu1
xalloys from a

consideration of their spin dynamics characteristics. The numerical results show that in dilute cobalt concentrations (i.e. xp0:4), the critical temperatures vary linearly with x for different itinerant carrier concentration conditions. Interestingly, the carrier concentration does not affect the degree of dependency of the temperature on the cobalt concentration when the carrier concentration is less than the atomic number concentration of the alloy.

r2006 Elsevier B.V. All rights reserved. PACS: 75.50.Cc; 75.20.Hr; 75.40.Gb

Keywords: Diluted magnetic alloy; s–d exchange model; Self-consistent Green’s function

1. Introduction

The magnetic properties of Co–Cu alloys have been intensively researched. The diverse magnetic phenomena exhibited by these alloys continue to attract attention because of their significant potential for magnetic recording applications. Early studies focused on the giant magne-toresistance (GMR) characteristics of an arrangement of alternating layers of magnetic and nonmagnetic layers

[1,2]. It is widely believed that GMR arises as a result of spin-dependent scattering [3]. Recently, many investiga-tions have been conducted into diluted nano-granular systems[4–7], in which a low concentration of magnetic Co particles is embedded in a nonmagnetic Cu host matrix. These systems provide a huge magnetoresistance effect, which greatly extends their range of potential application

[8]. It is evident that the magnetic interactions which take place in a diluted Co–Cu alloy play a key role in generating the GMR effect and therefore merit detailed study. In diluted systems, the magnetic interaction which takes place between Co ions with a local spin of S ¼3

2is believed to be

caused by the indirect exchange interaction through the

Ruderman–Kittel–Kasuya–Yosida (RKKY) mechanism

[9,10]. The minority magnetic ions are strongly hybridized with the majority itinerant electrons, thereby polarizing the electrons in the immediate surroundings. Under spin dynamics equilibrium conditions, over -polarization of the itinerant electrons takes place leading to the creation of an oscillatory magnetic coupling between the magnetic ions. Although clearly an essential consideration, spin dynamics are sometimes ignored in mean field theories. Consequently, this study uses the Green’s function method, which does take spin dynamics into consideration, to calculate the temperature dependence of magnetization[11]

in dilute Co concentrations.

2. Theory

In order to capture the spin dynamics physics of the diluted Co–Cu alloy, this study makes direct use of the Green’s function method to describe the spin dynamics s–d exchange model. The Hamiltonian is given by

H ¼ H0þ X I J Z d3rdð3Þðr
RIÞSIsðrÞ, (1)

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0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.01.227

Corresponding author. Tel.: +886 7 5919472; fax: +886 7 5919357. E-mail address: [email protected] (S.-J. Sun).

where J is the strength of the exchange interaction and SI and s denote the impurity spin density

and the itinerant spin density, respectively. The conduction band of the itinerant electrons is described by H0. The

itinerant electrons are contributed both by the Co atoms and the Cu ions. The present study aims to deal appropriately with spatial fluctuations. Hence, it is assumed that the dispersion of the itinerant electrons is described by a simple parabolic band, H0¼

P2=2m _{with effective mass m}_{. This study ignores the}

spin–orbit coupling under the assumption that it does not affect the magnetic properties qualitatively in metallic alloys. To describe the spin dynamics properly, it is convenient to apply the retarded Green’s function, which is defined as

GijðtÞ ¼ hhSþðRi; tÞ; S
ðRj; 0Þii

¼
iyðtÞh½SþðRi; tÞ; S
1ðRj; 0Þi ð2Þ

with the impurity spins localizing in coordinates Riand Rj,

respectively.

A more direct numerical approach to solving the Green’s function is to employ the equation of motion to it in the Heisenberg picture. The equation of motion for the Green’s function given above involves more Green’s functions of higher orders. To compute the Green’s function exactly therefore involves an infinite number of coupled equations for all orders of the Green’s function. To truncate this infinite series to a more manageable level, this study employs a mean-field decomposition known as the Ran-dom Phase Approximation (RPA), which is suitable for metals with dense carriers. The equation of motion is then given by id dtGijðtÞ ¼ j
JhS Z_ihhsþ ðRiÞ; S
ðRj; 0Þii þJhsZiGijðtÞ, ð3Þ

where j is the time differential value of the step function yðtÞ and hSZi and hsZ_{i are the impurity spin}

and itinerant spin densities, respectively. It can be seen that there exists an additional Green’s function hhsþ_ðR

iÞ; S
ðRj; 0Þii, which represents the itinerant spins

mediate the local spins. Employing the equation of motion for the additional Green’s function FijðtÞ in

mom-entum and energy representations leads to the following relationship:

ðO þ ekþq
ek
JhSZiÞF ðk þ q; k; OÞ

¼J

2ðf#ðekÞ
f"ðekþqÞÞGðq; OÞ. ð4Þ

Combining Eq. (4) and the Fourier transformation of Eq. (3) yields the following closed form of the Green’s function: O
JhsZi
JD 2V X k f"ðekÞ
f#ðekþqÞ O þ D þ ek
ekþqþiZ ! Gðq; OÞ ¼ j. (5)

The bosonic-like spin-wave dispersion is determined by identifying the pole in the Green’s function, i.e.

O
JhsZi
JD 2V X k f"ðekÞ
f#ðekþqÞ O þ D þ ek
ekþqþiZ ¼0, (6) where f_";#ðkÞ ¼ ½ebðekD=2
mÞ_þ1
1 _{is the Fermi}

distribu-tion of the itinerant carriers. The Zeeman gap, D  JhSZi, in the electronic band structure is due to the ferromagnetic order of the impurity spin and must be determined self-consistently. The impurity spin density is related to the spin-wave dispersion via Callen’s formula[12], i.e. 1 chS Z_{i ¼S
hn} swi þ ð2S þ 1Þhnswi2Sþ1 ð1 þ hnswiÞ2Sþ1
hnswi2Sþ1 , (7) where hnswi ¼ ½ebOðkÞ
1
1 is the average number of spin

waves.

As the temperature approaches the critical regime, the spin waves become denser and the kinematics constraint becomes significant. Solving Eqs. (6) and (7), the spin-wave dispersion is obtained self-consistently, and the averaged impurity spin density hSZi can be derived. The dispersion from Eq. (6) has two branches as a result of the presence of both localized itinerant spin densities. However, since the low energy fluctuations dominate, the optical branch can be ignored. Additionally, the spectral weight of the optical mode is small due to the dilute density of the itinerant carriers.

To complete the theoretical calculation, it is necessary to supply two physics quantities, namely the exchange coupling, J, and the itinerant carriers’ effective mass, m_.

This study assigns values of J ¼ 0:058 eV nm3 _{[13] and}

m_¼1:8m

e, where meis the mass of the bare electron. Note that this study deliberately specifies an overestimated value of m_{in order to reduce the Fermi-surface. In this way, the}

accuracy of the numerical integral calculation for the itinerant electrons band is increased and the qualitative behavior more fully revealed. Co and Cu are both metallic elements and both provide itinerant electrons to couple the local magnetic spins. In order to investigate the influence of the ratio of the itinerant electrons density to the density of the total metallic atoms, c/n, on the magnetic properties of the Co–Cu alloy, this study calculates the temperature dependence of the alloy’s magnetization at different c/n ratios.

3. Result and discussion

As shown inFig. 1, the magnetization is more robust at a higher concentration of embedded Co. Additionally, it is observed that the magnetization drops rapidly following the first-order transition near the critical temperature (TC). The current magnetization curves are different from the results from the general Weis-mean field theory near TC obtained by the current authors in a previous study [13]. The rapid drop in magnetization is thought to be the result of a magnon–electron decoupling, in which the spin

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dynamic equilibrium for local spin over-polarization through the itinerant electrons breaks down when the temperature reaches a particular temperature. As men-tioned previously, the RKKY plays through the local spin over-polarization under spin dynamic equilibrium condi-tions, which means that the density of the itinerant electrons and the separation distance between local impurities are significant inter-related issues. The sharp drop in magnetization predicted numerically in the present study is consistent with the experimental findings [14].

Furthermore, we will study the magnetic properties near TC to realize the magnetization sharp drop in detail. The dispersion of spin wave derived from the conventional spinwave theories[15]is temperature independent, which is an intrinsic characteristic for many kinds of magnets.Fig. 2

exhibits the dispersion of magnon from our theoretical calculation showing a temperature independent dispersion at far from TCs, which reveals a result that the temperature independence of normal spin wave existing at robust magnetism region, meanwhile it reveals an obvious magnon softening effect in the vicinity of TC leading to the magnetization falling down sharply. Interestingly, this softening effect starts from small magnon momentum qs then extending to whole dispersion region eventually. From the conventional spin wave theory as the q5, the magnon dispersion relation has O ¼ Dq2_{, where the stiffness}

constant D / J0 _{and J}0 _{is the magnetic coupling integral}

between two separated spins. From the linear response theory we have derived[16]before, the coupling J0_{is / J}2_,

where J is the coupling between itinerant spins and local spins. Therefore this softening effect results in D decreas-ing, which reveals a fact that the effective coupling J reduces and gives an implication with magnon–electron decouple in system.

Fig. 3shows that the Curie temperatures, TC, are linearly proportional to the Co concentration, x, for different values of the c/n ratio. These numerical results are again consistent with the experimental findings [14]. It can be

seen that the linear lines corresponding to different c/n ratios have different gradients, and intersect the horizontal (Co concentration) axis at different points. The results imply that there exists a particular value of the Co concentration at which the value of the Curie temperature is not affected by itinerant electrons donation. Further-more, the lines’ extrapolations do not pass through the zero point, which means that ferromagnetic states exist only above certain Co concentrations. Moreover, the Curie temperature is higher for higher c/n ratios above this concentration threshold, but lower for higher c/n ratios below this threshold. Experimentally [14], it has been shown that if the Co concentration is too low, then long range order ferromagnetic states will not exist. Finally,

Fig. 2shows that the linear lines are virtually the same at

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Fig. 1. Temperature dependence of magnetization as a function of Co concentration for constant c=n ¼ 1.

6x10-3 5x10-3 4x10-3 3x10-3 2x10-3 1x10-3 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 q Ω T = 30 K T = 42 K T = 40 K T = 44 K T = 20 K 2.7 2.4 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 Sz 0 5 10 Temperature (K) 15 20 25 30 35 40 45 50

Fig. 2. We take the exchange coupling and the effective mass are fixed at typical values J ¼ 0:15 eV nm3_{, m}_{¼0:5meand the ratio of itinerant and} localized spin densities fixed at c_{=c ¼ 0:1 to calculation resulting in} TC¼45 K. The inset of the figure shows a sharp drop of magnetization in the vicinity of TC. In the sharp drop region the magnon shows a softening

effect. 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0 TC 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 x CoxCu1-x c/n=1.0 c/n=1.5 c/n=2.0

Fig. 3. Variation of Curie temperatures with Co concentration for different ratios of itinerant carrier concentrations to total metallic ions, c/n.

S.-J. Sun et al. / Journal of Magnetism and Magnetic Materials 305 (2006) 428–431 430

low values of the c/n ratio, but diverge at higher values. This suggests that in CoxCu1
x alloys, the variation of TC

caused by the itinerant electrons density is limited to a finite range.

4. Conclusion

In conclusion, the theoretical approach presented in this study successfully captures the characteristic rapid reduc-tion in magnetizareduc-tion near the Curie temperature, TC. The

value of TCfor the CoxCu1
x alloy is found to be linearly

proportional to the magnetic Co concentration, x. Finally, the appearance of long range order ferromagnetic states occurs only above certain Co concentrations.

Acknowledgments

The authors gratefully acknowledge the assistance provided throughout this study by the National Center of Theoretical Science of Taiwan (SJS) and the financial support received by the National Science Council of Taiwan under Grant no. NSC 92-2212-E-110-030 (SPJ).

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