Concentration dependence of giant magnetoresistance in magnetic granular composites

Concentration dependence of giant magnetoresistance in magnetic granular composites

20 September 1999 Physics Letters A 260 Ž1999. 381–385 www.elsevier.nlrlocaterphysleta Concentration dependence of giant magnetoresistance in magnet...

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20 September 1999

Physics Letters A 260 Ž1999. 381–385 www.elsevier.nlrlocaterphysleta

Concentration dependence of giant magnetoresistance in magnetic granular composites Chen Xu a b

b,)

, Zhen-ya Li

a,b

CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China Department of Physics, Suzhou UniÕersity, Suzhou, 215006, China 1 Received 12 April 1999; accepted 4 August 1999 Communicated by A.R. Bishop

Abstract We combine effective medium theory ŽEMT. with the two-channel conducting model to study the magnetic granular concentration dependence of a giant magnetoresistance ŽGMR. in magnetic granular composites. The composite is composed of small magnetic granules Žsuch as Co. embedded in an immiscible nonmagnetic metallic matrix Žsuch as Ag.. We present a model for the composite in which the magnetic metallic granules are spherical in shape and have a distribution in sizes, and in which there are different contributions of superparamagnetic and ferromagnetic granules to conductance. The calculated result about the concentration dependence of GMR is in agreement with the experimental data. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction Recently, the magnetic granular concentration dependence of GMR in granular composites such as Co–Ag, Ni–Ag, Co–Cu granular films, has been researched w1–5x. It was shown that at a certain temperature the GMR effect increased as the magnetic granular concentration increased and after reaching a maximum the GMR effect decreased with the further increase of the concentration. There was a little theoretical work to study this phenomenon. Some theoretical work successfully explained the dependence of GMR on the granular size w6–9x, and a few papers discussed the magnetic granular con)

Corresponding author. Fax: q86-512-511259; e-mail: [email protected] 1 Mailing address.

centration dependence of the GMR w6x, but their calculation failed to explain the experimental results w1,4x. How to understand the concentration dependence of the GMR in these magnetic composites is the goal of this paper. In this paper, we will present a phenomenological theory to study the concentration dependence of GMR in Co–Ag granular. Firstly, based on the experimental data which have been reported in Ref. w5x, we assume that the composite contains small magnetic granules embedded randomly in an immiscible nonmagnetic metallic matrix and the granules have spherical shapes Žfor example, experiments showed the Co granules were spherical in shape w16x.. These magnetic granules have a distribution in sizes. Secondly, for small magnetic granules, if the temperature is higher than the average blocking temperature of the composite, most of these magnetic

0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 5 0 - 2

382

C. Xu, Z.-y. Li r Physics Letters A 260 (1999) 381–385

granules become superparamagnetic ŽSPM.. While the temperature is lower than the average blocking temperature, most of these granules become ferromagnetic Žand hence non-SPM. and others remain still SPM. We should distinguish the SPM from non-SPM granules, because they have a different contribution to the GMR w10–12x. Thirdly, we suppose that the conduction electrons in the composite are scattered Ž1. in the nonmagnetic matrix, Ž2. at the interface between nonmagnetic matrix and magnetic granular, and Ž3. within the magnetic spherical granules. 2. Theory Now, we assume the measure temperature is lower than the blocking temperature of the composite and the concentration of magnetic granules is p. Then we take a phenomenological parameter xs to describe the ratio of the SPM granule concentration to magnetic granule concentration. So the ferromagnetic granules concentration is pŽ1 y xs .. It is generally thought that the GMR is caused by heterogeneous magnetic structures. When a magnetic field B is applied to the magnetic granular composite, the magnetic moments of non-SPM granules mn are aligned and saturated, and the magnetic moments of SPM granules ms have angles Ž u . of intersection with applied field B. We adopt two-channel conducting model and effective medium theory ŽEMT. to study the magnetic granule concentration dependence of GMR w9x. For a non-SPM ferromagnetic granule, the spin direction of conduction electrons may be parallel Žsay ‘‘spin-up’’. or anti-parallel Žsay ‘‘spin-down’’. to the direction of the magnetic moment of the granule. For a spin up electron, the contribution to its resistance will be high due to scattering events and then to its conductance will be low ŽL., while for a spin-down electron, the contribution to its conductance will be high ŽH.. Let sq and sy indicate the conductivities for spin-up and spin-down electrons respectively. Using the signs ‘n’ Žor ‘s’. for non-SPM Žor SPM. granules, we describe the conductivities of non-SPM and SPM granules for spin-up or spin-down electron as sn" and ss " , respectively. We assume that the magnetic moments of nonSPM ferromagnetic granules will be quickly aligned

and saturated in the presence of an applied field B Žsuch as B s 5.0 T.. Then, for a given concentration of magnetic granules, the average conductivities ² sn": can be expressed by q ² snq: s snL ,

Ž 1. Ž 2.

y ² sny: s snH ,

q y where snL and snH are independent of B. " But for the ss , the magnetic moments msŽ B . of SPM particles are dependent on B. We presume the values of ss " change linearly with cos u Žwhere u is the angle of intersection between ms and B . according to the proposal of Gittleman et al. w13x. Therefore we get

ssq s 12 Csq ms Ž B . Ž 1 q cos u . q 21 Cy s ms Ž B . Ž 1 y cos u . ,

ssy s 12 Csy ms q

1 2

Ž 3.

Ž B . Ž 1 q cos u . Ž B . Ž 1 y cos u . ,

Cq s ms

Ž 4.

where Cs" are two coefficients. After considering the distribution of SPM moments, we get the thermal average of ss " as follows w12x: ² ssq : s 12 Ž ssL q ssH . I1 Ž B . q 12 Ž ssL y ssH . I2 Ž B . ,

Ž 5. ² ssy : s 12 Ž ssL q ssH . I1 Ž B . q 12 Ž ssH y ssL . I2 Ž B . ,

Ž 6. where I1 Ž B . s

1

ln w sinh a m ra m x ,

am

I2 Ž B . s y

LŽ am .

am

1 q

am

am

H0

Ž 7.

ž

1y

2 LŽ a .

a

/

da ,

Ž 8. am s

mm B k BT

,

Ž 9.

and ssL and ssH are two coefficients, LŽ a . is the Langevin function, mm is the largest magnetic moment of the Co granule which is unblocked SPM particle at temperature T. As the magnetic granules are randomly embedded with concentration p in the nonmagnetic matrix, we use effective medium theory ŽEMT. w14,15x to get the average conductivities seq and sey for spin-up

C. Xu, Z.-y. Li r Physics Letters A 260 (1999) 381–385

and spin-down conduction electrons, respectively. Then, the average conductivity se " can be obtained from following equations: ² snq: y seq ² ssq: y seq p xs q q p 1 y x Ž . s ² ss : q 2 seq ² snq : q 2 seq qŽ 1 y p . p xs

s h y seq s h q 2 seq

² ssy: y sey ² ssy : q 2 sey

qŽ 1 y p .

s 0,

q p Ž 1 y xs .

s h y sey s h q 2 sey

s 0,

Ž 10 . ² sny: y sey ² sny : q 2 sey

Ž 11 .

where s h is the conductivity of the Ag matrix. Being used the two-channel conducting model, the composite’s effective conductivity se are

se s seqq sey.

Ž 12 . Ž . We have the magnetoresistance MR of the composite according to the definition of MR as follows:

MR s

sey1 Ž B . y sy1 Ž 0 . sey1 Ž B .

.

Ž 13 .

In order to compare with the experimental data of magnetic granules concentration dependence of GMR in Ref. w5x, we consider two cases: measure temperatures at T s 4.2 K and T s 300 K with the same applied field B s 5.0 T. Ž1. T s 4.2 K Ž B s 5.0 T.. The blocking temperature of Co–Ag composite sample is about 20 K at low concentration of Co granules according to Ref. w5x. And when the Co concentration increases, the blocking temperature will also increase. Because the blocking temperature is much higher than the measure temperature, we can assume that the non-SPM granule concentration is much larger than SPM granules. The phenomenological ratio xs of SPM granule concentration to whole Co concentration is small and its value decreases when the Co concentration increases. Because of the number of the Co granules is different with the change of concentration, we assume the phenomenological parameter gn Žor gs ., to describe the probability and degree of scattering on the interface between nonmagnetic and non-SPM Žor SPM. as well as within the non-SPM Žor SPM. granules. Then, the average conductivities ² sn": of non-SPM granules which is in majority can be

383

rewritten as ² snq: s gn sq L ,

Ž 14 . Ž 15 .

² sny: s gn sy H.

It is implied that the average conductivities ² sn": are changed with the increase of Co concentration. As the Co concentration is low, the magnetic granules are separated and the scattering number will increase when the number of magnetic granules increases. However, when the Co concentration is about 20%, the small magnetic granules begin to interconnect and fold up, thus the effective surface for scattering decreases. We can assume gn is maximum as 1.0 for the largest limit of separated granules at concentration 19% w5x according to the experimental data. Putting the Eqs. Ž14. and Ž15. into Eqs. Ž10. and Ž11. respectively, we will obtain ² se ": when parameters are given. Now, we select the values of conductivities as 4.16 for Ag and 1.0 Žarbitrary units. y for Co, 0.01 for sq L and 10.0 for s H . For SPM granules, we neglect the influence of gs and take ssL as 0.1and ssH as 2.0 because of the very small SPM concentration and assume the maximum moment ms about 4.5m B as the sizes of SPM granules are in nanometers. The two phenomenological parameters xs and gn will be determined by a nonlinear leastsquare fit to the GMR data in Ref. w5x. In Table 1, we show the values of gn and xs for various Co concentrations. The selection of gn and xs is reasonable for above assumptions. Namely, the value of gn increases first then decreases later as Co concentration increases and xs decreases as Co concentration increases. The curve of GMR vs concentration at T s 4.2 K is shown in Fig. 1 Žthe filled dots are experimental data.. The theoretical result is in agreement with the experimental report in Ref. w5x.

Table 1 Values of the parameters at 50 K Vol.Ž%.

7.14 18.83 23.70 35.23 49.03

T s 4.2 K

T s 300 K

gn

xs

gs

0.6640 0.9480 0.7250 0.4340 0.3746

0.070 0.060 0.058 0.040 0.020

0.5250 0.6560 0.5720 0.4115 0.3608

C. Xu, Z.-y. Li r Physics Letters A 260 (1999) 381–385

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higher ssH and lower ssL . The value of parameter gs is given in Table 1 and ² se ": can be obtained by p

p

² ssq: y seq ² ssq : q 2 seq ² ssy: y sey ² ssy : q 2 sey

q Ž1 yp.

q Ž1 yp.

s h y seq s h q 2 seq s h y sey s h q 2 sey

s 0,

Ž 18 .

s 0.

Ž 19 .

The theoretical curve is in good agreement with experimental data Žopen dots. in Fig. 1.

3. Discussion and conclusion Fig. 1. Magnetic Co concentration dependence of the GMR at two different temperatures. The symbols represent the experiment data of Ref. w5x and the curves give the calculated values.

Ž2. T s 300 K Ž B s 5.0 T.. The average blocking temperature Ž20 K. of the composite sample is much lower than the measure temperature 300 K at the range of 0 to 50% Co concentration according to the experimental results in Ref. w5x. So we can assume the Co–Ag composite containing only SPM granules. The influence of parameter gs can not be neglected because in this case, all the magnetic granules are superparamagnetic. And we can predict that gs is smaller than gn at the same Co concentration because a granule’s magnetic moment is smaller at SPM state than in non-SPM state. Namely, nonSPM states are more effective on spin-dependent conductance, so gn should be larger than gs . Thus, we should substitute gs ssL and gs ssH for ssL and ssH in Eqs. Ž3. and Ž4.. Then ² ss ": can be rewritten by ² ssq : s 12 gs Ž ssL q ssH . I1 Ž B . q 12 gs Ž ssL y ssH . I2 Ž B . , ² ssy : s 21 gs

Ž 16 .

Ž ssL q ssH . I1 Ž B .

1 2

q gs Ž ssH y ssL . I2 Ž B . .

Ž 17 .

Now we take ssL as 0.01 and ssH as 10.0, and assume the maximum moment ms as 744m B , which is much larger than the value of ms in case 1 at low temperature because the maximum size of the Co granule is much larger than that in case 1. The maximum moment is larger and correspondingly

We have studied the magnetic granule ŽCo. concentration dependence of GMR by a phenomenological method. In order to study the magnetoresistance of the composite which contains fine magnetic granules, we should distinguish different contribution to spin-dependent conductance between superparamagnetic and ferromagnetic granules. The blocking temperature is important to determine whether the Co granules lie in SPM state or in mixed non-SPM and SPM state. And we can only give a phenomenological parameter xs to describe the concentration of SPM and non-SPM state. When given the magnetic Co concentration, the parameters gn, s reflect the spin-dependent scattering at the magnetic-nonmagnetic interface and within the magnetic granules. The surface spin-dependent scattering is more important than body scattering within magnetic granules. Therefore gn and gs increase at first with the increase of Co concentration then begin to decrease at the concentration about 20%. We studied the GMR effect at two measure temperatures 4.2 K and 300 K when the magnetic Co concentration was changed. Using our method, we predict that when measure temperatures are between 4.2 K and 300 K, the peaks of GMR will also be between the two peaks in Fig. 1. The main reason is that the non-SPM granule concentration will decrease when temperature increases from 4.2 K to 300 K, which lead to smaller gn and larger xs . In summary, we combined effective medium theory with two-channel conducting model to calculate the magnetic Co concentration dependence of the GMR. The result is in good agreement with experi-

C. Xu, Z.-y. Li r Physics Letters A 260 (1999) 381–385

ment, this shows the model presented and the theoretical method are essentially applicable for magnetic granular composites. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant 19774042. References w1x C.L. Chien, J.Q. Xiao, J. Samuel Jiang, J. Appl. Phys. 73 Ž1993. 5309. w2x Shi-hui Ge, Ying-Yang Lu, ¨ Zong-Zhi Zhang, Cheng-Xian Li, Tao Xu, Jia-Zheng Zhao, J. Magn. Magn. Mater. 168 Ž1997. 35. w3x A. Gerber, A. Milner, B. Groisman, M. Karpovsky, A. Gladkikh, Phys. Rev. B 55 Ž1997. 6446.

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