26 July 1999
Physics Letters A 258 Ž1999. 401–405 www.elsevier.nlrlocaterphysleta
A phenomenological theory of giant magnetoresistance in magnetic granular alloys Chen Xu a , Zhen-Ya Li a b
Department of Physics, Suzhou UniÕersity, Suzhou 215006, China CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China Received 5 February 1999; accepted 27 May 1999 Communicated by J. Flouquet
Abstract We combined the effective medium theory ŽEMT. with the two-channels conducting model to study the giant magnetoresistance ŽGMR. effect of magnetic granular alloys. These fine magnetic granules were randomly embedded in an immiscible nonmagnetic metallic matrix. It is assumed that these magnetic granules with concentric coated shells are spherical in shape and have a distribution within a rage of sizes. The volume fraction of magnetic granules and the annealing temperature dependencies of GMR were investigated by use of a phenomenological theoretical method, the results of our calculations agreed with experimental data. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Since the giant magnetoresistance ŽGMR. effect was first discovered in magnetic multilayers w1x and subsequently in heterogeneous alloys w2,3x, numerous studies on magnetic alloys have been reported Žsee Refs. w4,5x for recent reviews.. Most of these heterogeneous alloys are composed of magnetic granules embedded randomly in an immiscible nonmagnetic metal. Recently, some experiments with such alloys showed that there was a GMR peak value at a volume fraction 15–25% of the magnetic component, near the percolation threshold w6–9x. The annealing temperature dependence of GMR was also reported w7,10,11x. There are a few theoretical studies on GMR vs. magnetic granular volume fraction
Corresponding author, at Suzhou University.
or GMR vs. annealing temperature w7,12,13x. Rubinstein w12x extended Valet and Fert’s model w14x and he concluded that the resistivity r and D r decrease monotonically with annealing temperature Ta , so that the behavior of GMR cannot be explained. Chang and Lo w13x extended Rubinstein’s theory to include mutual interactions between the granules and derive an analytical solution for a granular system. They succeeded in explaining the influence of particle size on GMR. However, their calculated behavior of GMR vs. volume fraction is quite different from experimental data w6,7x. In this Letter, we consider a binary alloy consisting of fine magnetic granules embedded randomly in an immiscible nonmagnetic metallic matrix. These small magnetic granules can be assumed as spherical in shape from experimental observation w15x. These granules may also be coated with shells, the reasons are: Ž1. The immiscible system makes it possible to
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C. Xu, Z.-Y. Li r Physics Letters A 258 (1999) 401–405
fabricate phase-separated heterogeneous alloys with ferromagnetic granules embedded in a nonmagnetic matrix. Phase separation is essential to the GMR effect while the system which is a solid solution Žsuch as Fe–Pt. does not exhibit GMR w7x. Ž2. The magnetic moments of granules play an important role in the GMR, experiments showed that the surface atoms attributed to the magnetic moments were different from those within the granules w16x. We assume the granules with shell structure in order to distinguish the difference between spin-dependent scattering at the magnetic-normal interface and within the magnetic granules. In fact, the radii of the fine magnetic granules are varied within a range of sizes. Therefore, these granules may be superparamagnetic, single-domain and multi-domain according to their sizes w17x. We called single-domain granules as ferromagnetic ŽFM., and superparamagnetic and multidomain granules as non-ferromagnetic Žnon-FM.. In general, as-sputtered samples are properly annealed, the fine magnetic granules will be larger and some of them may exhibit multi-domain structure. We combined effective medium theory ŽEMT. w18,19x with the two-channels conducting model to study the magnetic granular concentration and annealing temperature dependencies of GMR. The calculation is in agreement with the experimental results. 2. Theory First, we consider a spherical magnetic granule with a magnetic shell embedded in an immiscible nonmagnetic matrix. We assume the radii of core and shell axe rc and rs , respectively. sc , ss and sm are the conductivities of the concentric core, shell and matrix, respectively. We can simplify this threecomponent composite as an equivalent two-component composite with a solid sphere embedded in the same matrix w20x. The conductivity s˜ of the solid sphere is
s˜ s msc ms
x Ž 1 q 2 x . q 2 px Ž 1 y x .
Ž1 q2 x . ypŽ1 yx .
Ž 1. Ž 2.
where the ratio x ' ssrsc and a structural parameter p ' Ž rcrrs . 3. The radius of the solid sphere is rs .
Now when these coated spherical magnetic granules are randomly embedded in a metallic matrix, we assume the volume fraction of the cores is f c . Actually, the radii of these magnetic granules have a distribution within a range of sizes. At a given temperature, some of the magnetic granules are ferromagnetic ŽFM. and others are paramagnetic ŽnonFM.. Then we take the volume fraction of FM and non-FM granules as f n and fs , respectively. The relation f n q fs s f c should be satisfied. When magnetic field B is applied to the alloys, we assume all the magnetic moments of FM granules will be aligned along the direction of B and the moments of non-FM granules will have varied angles with B. The FM and non-FM granules have different effect on spin-dependent conductance electrons w21,22x. The spin direction of conductance electrons may be parallel Ž‘‘spin-up’’. or anti-parallel Ž‘‘spin-down’’. to the direction of the magnetic moment of initial granules. We define that for spin-up electrons the contribution to its conductance will be low due to scattering events and for spin-down electrons will be high. Let sq and sy indicate the conductivities for spin-up and spin down electrons, respectively. Using the subscripts ‘c’ for cores and ‘n’ Žor ‘s’. for FM Žor non-FM. granules, we describe the average conductivities of FM and non-FM granular cores for spin-up and spin-down electrons as snc" and ssc" , respectively. For simplicity, we assume all the magnetic granules coated with shells have the same structural parameter p. Therefore, we can use equivalent solid spheres instead of these coated spheres. The corresponding equivalent conductivities s˜n" ,s and volume fraction f˜n",s of these solid spheres can be written as follows
s˜n" s msnc" s˜s " s mssc" f˜n",s s
f n ,s p
Ž 3. Ž 4.
where m is given by Eq. Ž2., and f n,s is the volume fraction of the magnetic spherical cores. Based on the two-channels conducting model, we calculate the effective conductivities seq and seq for spin-up and
C. Xu, Z.-Y. Li r Physics Letters A 258 (1999) 401–405
spin-down electrons, respectively, by using EMT as follows f˜s
s˜sq y seq s˜sq q 2 seq
s˜nq y seq s˜nq q 2 seq
sm y seq sm q 2 seq
Ž 5. f˜s
s˜sy y sey s˜sy q 2 sey
s˜ny y sey s˜ny q 2 sey
sm y sey sm q 2 sey
Ž 6. where sm and fm indicate the conductivity and the volume fraction of the matrix, respectively. Then, when the magnetic field is applied, the effective conductivity of the alloy can be obtained as
se Ž B . s seq q sey
Then the magnetoresistance ŽMR. of the alloy can be represented as follows, MR s
se Ž B . y se Ž 0 . se Ž B .
where seŽ0. is the effective conductivity of demagnetized alloy.
3. Comparison with experimental data Eq. Ž5. and Eq. Ž6. are used to calculate the GMR vs. magnetic volume fraction for the Co–Ag alloy sample. We select the conductivities of Co and Ag as 1.0 and 4.16 Žarbitrary units.. We assume the magnetic Co granules coated with thin shells and consider the magnetic surface spin-dependent scattering to be more important than bulk scattering within the magnetic granules. Then, we choose the structural parameter p s 0.9 Ža thin shell.. The ratios of conductivity of shells to conductivity of cores for FM and non-FM granules are selected as follows q sns q snc y sns y snc
s 0.2 s 0.2
Ž 9. Ž 10 .
where sns" and snc" are the conductivities of FM shells and cores while sss" and ssc" are the conductivities of non-FM shells and cores, respectively.
y q q rsnc and sscyrssc are given at a certain If snc measuring temperature, we can obtain the ratios s˜nyrs˜nq and s˜syrs˜sq of the equivalent solid spheres by using Eq. Ž3.. We introduce a parameter fsrfn which changes when the magnetic volume fraction increases. Because when magnetic concentration increases, the diameters of magnetic granules may also be enlarged. So there is an optimum range of sizes for magnetic granules being ferromagnetic w17x. If the diameters of the magnetic granules are smaller or larger than the optimum range, these granules will be superparamagnetic or multi-domain. We therefore predict that the ratio fsrfn decreases first, and then increases when volume fraction f c increases. Namely, f nrfc will have a maximum value with increasing f c . Due to the fact that ferromagnetic granules have greater effect on GMR than superparay q magnetic granules, we select snc rsnc s 40.0 and y q ssc rssc s 3.4 at a high measuring temperature of 300 K in order to compare the results with the experimental date w9x. The ratio fs to f n versus varying magnetic concentration is listed in Table 1. fsrfn data are obtained based on the experimental data at 300 K. The ratio fsrfn decreased first then increased as we predicted. We use a nonlinear leastsquares method to determine the relation between magnetic concentration and fsrfn . Then we attempt to fit to the curve of GMR vs. magnetic concentration at a low measuring temperature of 4.2 K. For the measuring temperature has different effect on FM and non-FM magnetic granules when the magy q q netic field is applied, the ratios snc rsnc and sscyrssc will change at different measuring temperatures. We y q then select snc rsnc s 57 and sscy rsscq s 4.8 Žat 4.2 K. which are larger than those at 300 K and obtain the GMR vs. magnetic concentration curve shown in Fig. 1 using the same fsrfn data as in Table 1. The
Table 1 Values of fs r fn change with the increase of magnetic concentration f c at a fixed measuring temperature of 300 K f c Ž=10y2 .
fs r f n
7.14 1.88 2.37 3.52 4.90
0.960 0.256 0.525 1.850 3.430
C. Xu, Z.-Y. Li r Physics Letters A 258 (1999) 401–405
Fig. 1. Magnetic granular concentration dependence of the GMR in a Co–Ag alloy. The symbols represent the experimental data and the curves give the calculated values.
calculated curve is in agreement with experimental data. Using our method, we can also explain the dependence of GMR on annealing temperature. Thermal annealing will enlarge the sizes of granules w7x, and the sizes of granules increase with increasing annealing temperature. So an optimum range of sizes may also exist for the FM state at a proper annealing temperature. For an Fe 20 Ag 80 sample w7x, its magnetic volume concentration f c is 20%, we choose y q s Fe s 1.0 and sAg s 5.8 Žarbitrary units., snc rsnc s y q 40 and ssc rssc s 4.5. We obtained the f nrfs ratio at the measuring temperature of 300 K according to the experimental data shown in Table 2. It is obvious that there is a proper annealing temperature which lead to a maximum GMR. We still noticed that the y q q rsnc and sscyrssc varied with measuring ratios snc y q temperature. Then we chose snc rsnc s 65 and sscyr Table 2 Values of f n r fs change with the increase of annealing temperature at a fixed measuring temperature of 300 K Ta Ž8C.
f n r fs
As Ž ; 50. 200 250 300 350 400 450
1.160 1.320 0.935 0.545 0.294 0.074 0.0184
6.59 6.99 5.89 4.22 2.63 0.70 0.09
Fig. 2. Annealing temperature dependence of the GMR in an Fe 20 Ag 80 sample. The symbols represent the experimental data and the curves give the calculated values. y q q sscq s 5.6 for 77 K, snc rsnc s 103 and sscyrssc s 6.2 for 4.2 K to fit the experimental data, and the f nrfs data are the same as in Table 2. The symbols in Fig. 2 are experimental data and the curves were our calculation results. It is in good agreement with the experimental results.
4. Discussion and conclusions In this Letter, we have studied spin-dependent scattering in the magnetic granular alloys and used a thin shell structure to reflect the interfacial effect. We considered the difference of conductivity between the thin shells and cores when a magnetic field was applied to it. If the ratio of ss Žshell. to sc Žcore. enhances, the GMR will increase. In other words, if the interfaces between magnetic granules and nonmagnetic matrix have stronger influence on conductance electrons than within magnetic granules, the GMR effect will be enhanced from our model. We introduced a phenomenological parameter fsrfn Žor f nrfs . to describe the ratio of concentration of FM granules to that of non-FM. The ratio fsrfn changed with increasing magnetic granular concentration or annealing temperature. According to the data listed in Tables 1 and 2, we can learn that there is an optimum value of fsrfn causing a maximum GMR when magnetic concentration or anneal temperature increases. It indicates that the distribution of a range of sizes exists in the magnetic granular
C. Xu, Z.-Y. Li r Physics Letters A 258 (1999) 401–405
alloys, a portion of these granules is FM and get to the maximum at a proper concentration or annealing temperature. For FM granules are more contributive to GMR than non-FM granules, the GMR will reach a peak when the system contains the maximum ratio of f nrfc . Having noticed the conductivities of magnetic granules varied with measuring temperature, we calculate the GMR at different measuring temperatures by using fsrfn data which were determined by fitting to GMR at measuring temperature 300 K. The GMR vs. magnetic volume fraction or annealing temperature is shown in Fig. 1 or Fig. 2, respectively. The curves in agreement with experimental data are obvious. In summary, we combined the effective medium theory with the two-channels conducting model to calculate the dependence of GMR on magnetic volume fraction or annealing temperature. The results are quite well in agreement with experimental data. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant 19774042. References w1x M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Crenzet, A. Friederich, J. Chazelas, Phys. Rev. Lett. 61 Ž1998. 2472.
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