- Email: [email protected]
The study on the giant magnetoresistance in magnetic granular alloys
27 November 2000
Physics Letters A 277 (2000) 169–174 www.elsevier.nl/locate/pla
The study on the giant magnetoresistance in magnetic granular alloys Yan Ju a , Zhenya Li a,b,∗ a Department of Physics, Suzhou University, Suzhou 215006, China b CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China
Received 2 August 2000; accepted 26 October 2000 Communicated by J. Flouquet
Abstract To study the giant magnetoresistance in magnetic granular alloys, an effective cluster model is extended to include spindependent interface scattering and bulk scattering. The dependence of giant magnetoresistance on particle size and volume fraction of magnetic component is discussed, some features of the giant magnetoresistance observed in recent experiments can be explained and the optimal choice of parameters to maximize the giant magnetoresistance can be determined. 2000 Elsevier Science B.V. All rights reserved. PACS: 75.70.Pa; 75.70.Ak; 75.70.-i; 75.50.Tt Keywords: Giant magnetoresistance; Granular composite; Effective cluster model
1. Introduction The giant magnetoresistance (GMR) in magnetic granular alloys has been studied thoroughly, since it provides a new perspective for technological applications. It is generally believed that GMR is mainly due to the spin-dependent interfacial scattering as well as the spin-dependent bulk scattering. However, there are still some aspect lacking good theoretical explanation, such as the volume fraction dependence of GMR in granular alloys. The relevant experimental reports [1–3] demonstrate some universal features: (1) The GMR reaches a maximum at the volume fraction of magnetic component p = 0.15–0.25 [1,2] or about 0.4 [3]. (2) As p further increases beyond the
* Corresponding author. Fax: +86-512-5112907.
E-mail address: [email protected] (Z. Li).
peak region, the GMR drops and GMR ' 0 at p → 1. These workers attributed it to the increasing coalescence of magnetic particles with the increment of p. The formed large multidomain clusters have no contribution to GMR [4], hence the largest GMR occurs below the percolation threshold of the system [3]. Recently, a few theoretical studies were developed on the GMR for the granular alloys. Some works [5,6] had respectively explained GMR vs. annealing temperature successfully, but had failed the GMR vs. volume fraction of magnetic component. Some works [7] made an attempt to include the mutual interactions, however, their model could not be applied to high volume fraction region, so the results was quite different from experimental observations. Within the framework of effective medium approximation (EMA), the peak of GMR was obtained [8,9], but there were still great discrepancy with the recent experimental data [1–3] even compared with their best results.
0375-9601/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 0 ) 0 0 7 0 9 - X
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Y. Ju, Z. Li / Physics Letters A 277 (2000) 169–174
EMA, which assumes the components distributed in highly random fashion, is a useful method to treat the transport properties of granular composites. Nevertheless, it seems to be quite unsatisfactory for investigation the effective properties of systems due to ignoring the realistic structures of the medium. Higher multipole interactions produce strong short range forces that drive similar particles clustering during sample preparation [10]. Based on the simple effective cluster model, Doyle [10] had got good theoretical results on the wavelength dependent optical transmission of granular CoSiO2 films over the entire range of metal volume loading. In this Letter, we extended the effective cluster model to research GMR in granular alloys, with the consideration of spin-dependent interface scattering and bulk scattering. Our results could well explain the observed dependence of GMR on size and volume fraction of the ferromagnetic component.
2. Formalism 2.1. The general expression for the conductivity in the effective cluster model We first consider the composite consists of two kinds of granules with conductivity σ1 and σ2 , present in volume fractions p and 1 − p, respectively. The granule 1 with σ1 exists both in the region of compact cluster and in the region of being isolated particle surrounded by granule 2 with σ2 . These two regions intermingle with each other and exactly fill the whole space. The volume fraction of the cluster region is V1 and the one of the uncluster region is V2 . Denoting the fraction of granule 1 in the cluster region by f , it is easy to know that V1 = fp
(1)
and V2 = 1 − fp.
(2)
The conductivity of cluster region σcluster and uncluster region σmg , treated as constituent conductivity, are inserted in EMA equation as (1 − fp)
σmg − σemt σcluster − σemt + fp = 0; (3) σmg + 2σemt σcluster + 2σemt
here, σemt is the effective conductivity of the composite which we search for. The fraction of granule 1 in the uncluster region is (1 − f )p. Therefore the actual volume fraction p0 of granule 1 in the uncluster region, relative to the volume of uncluster region, is p0 =
(1 − f )p (1 − f )p . = V1 1 − fp
(4)
Hence it is straightforward to derive the effective conductivity of the uncluster region by the Maxwell– Garnett equation σmg − σ2 (1 − f )p σ1 − σ2 = . σmg + 2σ2 1 − fp σ1 + 2σ2
(5)
2.2. The application to GMR in ferromagnetic alloys Let us consider a system composed ferromagnetic granules (such as Co, Ni, Fe) and nonmagnetic granules (such as Cu, Ag, Au), presenting in volume fraction p and 1 − p, respectively. The ferromagnetic granules exist in two regions: (1) cluster region with α , formed by compact ferromagnetic granules; σcluster α , being isolated single do(2) uncluster region with σmg main ferromagnetic particle surrounded by nonmagnetic granules. We define the superscript α = +(−) for the electron spin parallel (antiparallel) to the magnetic moment of ferromagnetic component. Coated with shells which are thin regions of mixed ferromagnetic and nonmagnetic atoms [11], the isolated particles could be regarded as equivalent solid particles [12] with σ˜ α : σ˜ α =
γ α (1 + 2γ α ) + 2λγ α (1 − γ α ) α σc ; (1 + 2γ α ) − λ(1 − γ α )
(6)
σα
a 3 ) . σcα and σsα are the here, γ α = σsα , λ = ( a+t c conductivities of core and shell of isolated particles, respectively. a is the radius of the core and t is the thickness of the shell. Taking the limit t → 0, the ratio γ α = t/σsα has a finite value [6,8,9] and σ˜ α is reduced to
σ˜ α =
a σ α. a + γ α σcα c
(7)
According to Eq. (5) we can get the effective conducα of the uncluster region, tivity σmg
Y. Ju, Z. Li / Physics Letters A 277 (2000) 169–174 α −σ σmg n α σmg
+ 2σn
=
(1 − f )p σ˜ α − σn , 1 − fp σ˜ α + 2σn
(8)
where σn is conductivity of nonmagnetic granule. The spins of electrons only have two directions, parallel and antiparallel to +z axis (such as the direc→ − tion of the applied magnetic field H ), hence electrons are divided into spin-up and spin-down channels correspondingly. The number of electrons in each chan− → → − nel is equal without H . Applied strong H , the two channels become uneven, for electrons skipping from spin-down channel to spin-up channel. It is known [2] that conductivity depends on not only the scattering strength but also the number of conduction electrons per unit volume. Therefore, at present of applied → − field H , the conductivity of each entities in spin-up and spin-down channels should be revised correspond− → ingly, relative to the one without H , by multiplying by W1 and W2 which indicate the change of electron number in spin-up and spin-down channels, i.e., the spin-polarization of electrons. It is nature that W1 > 1 and W2 < 1. Magnetoresistance (MR) is defined as the difference of resistivity between the completely magnetized state and the completely demagnetized stated. For the magnetized state, all the magnetic components have same magnetization direction parallel to the strong applied − → field H (point up), so the effective conductivity of the composite is
171
magnetization pointed up and pointed down, so the effective conductivity σemt in each channel should be equal. It can be derived as + − −σ σmg − σemt 1 − fp σmg emt + + − 2 σmg + 2σemt σmg + 2σemt + − − σemt σcluster fp σcluster − σemt + = 0. + + − 2 σcluster + 2σemt σcluster + 2σemt (12) Hence the effective conductivity of σ (0) ≡ σ (H = 0) of the composite should be σ (0) = 2σemt .
(13)
Then the GMR of the composite can be represented as follows: σ (H ) − σ (0) . GMR = (14) σ (H ) 2.3. Discussion about the cluster fraction f
= 0,
(10)
= 0.
(11)
The fraction f of ferromagnetic component in clusters plays a critical role in the effective cluster model. In general, the cluster fraction should be a function f (p) of the relative volume loading of magnetic component. The proper choice of f depends upon the preparation of the composites. The magnetic alloys [1–3] were made by cosputtering or coevaporation methods that deposited the components in atomic dispersion. If sputtered atoms and molecules arriving at the surface layer during sample preparation have a relatively high surface mobility, they will form two-dimensional network in each sputtered layer, even though the film is of three dimensions [10]. Hence, the cluster fraction f should be in the form of two dimensions and it should be related with percolation threshold pc of the composite for the reason that it is easy to form cluster if pc is relatively low. Therefore, we assume 1 for p < 12 + pc , [1−(1−(1/2−pc+p))1/2 ]2 f = [1−(1−(1/2−p (15) 1/2 ]2 +[1−(1/2−p +p)1/2 ]2 c +p)) c for p > 12 + pc .
→ − For the demagnetized state, no field H is applied. The spin-up and spin-down channels are even and there are the same numbers of magnetic components with
In fact, it turns to just the function f2 in Ref. [10], when taking a characteristic two-dimensional percolation threshold pc = 0.5. Doyle has achieved good agreement with experiments for optical transmission
+ − + σemt . σ (H ) = σemt + σemt
(9)
− σemt
and are the effective conductivity for Here, spin-up and spin-down channels. According to Eq. (3), we can get, respectively, (1 − fp) + fp
+ + (W1 σmg ) + 2σemt + + (W1 σcluster ) − σemt
+ + (W1 σcluster ) + 2σemt
(1 − fp) + fp
+ ) − σ+ (W1 σmg emt
− ) − σ− (W2 σmg emt
− − (W2 σmg ) + 2σemt − − (W2 σcluster ) − σemt
− − (W2 σcluster ) + 2σemt
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Y. Ju, Z. Li / Physics Letters A 277 (2000) 169–174
measurements and conductivity measurements of Au– SiO2 samples, by adopting the function f2 [10]. On the other hand, the sputtered atoms and molecules will form three-dimensional network, if they have a relatively low surface mobility. We assume, in a similar way, cluster fraction f in a three-dimensional form as 1 for p < 12 + pc , [1−(1−(1/2−pc +p))1/3 ]3 f = [1−(1−(1/2−p (16) 1/3 ]3 +[1−(1/2−p +p)1/3 ]3 c +p)) c for p > 12 + pc .
3. Discussion In this Letter, the GMR effect for magnetic alloys is studied according to Eqs. (7)–(16) and the influence of volume fraction of magnetic component and the particle size on GMR are discussed. For the reason that the compact multidomain cluster have no contribution − + = σcluster = 1 (arbitrary to GMR [4], we select σcluster units). It is known that the conduction electrons experience strong scattering and hence result in low conductivity, if their spins are antiparallel to the magnetic moments of the ferromagnetic particles. Therefore, we take a value lower than σcluster for the conductivity σc− such as 0.8 in our calculation. Figs. 1–4 study the dependence of GMR on the volume fraction p of ferromagnetic contents. There is a peak of GMR in our calculations. The ferromagnetic components exist in the form of compact clusters and isolated particles. It is only the isolated particles that contribute to GMR. In the magnetically dilute region, ferromagnetic components mainly exist in the form of isolated particles, so GMR increases with p due to the increasing concentration of magnetic scattering centers. As p further increases, cluster fraction raises rapidly and GMR drops beyond the peak region. When p → 1, most of magnetic components coalesce into clusters, hence there are no GMR effect. We find that the position for the maximum of GMR mainly depends on the internal structures of films. In Fig. 1, we present GMR vs. volume fraction for a two-dimensional network with pc = 0.5 and for three-dimensional networks with pc = 1/3 and 0.2, respectively. The calculation results show that the largest GMR occurs at about p = 0.42, 0.26 and 0.18.
Fig. 1. The volume-fraction dependence of GMR: (a) for two-dimensional network with pc = 0.5, taking W1 = 1.4 and W2 = 0.6; (b) and (c) for three-dimensional networks with pc = 1/3 and pc = 0.2, taking W1 = 1.8 and W2 = 0.2.
Wang et al. [1] and Gerber et al. [3] studied GMR for CoAg alloys, respectively, and found the peak of GMR at about p = 0.25 and 0.40–0.42. Our results are in good agreement with their experimental reports. With different conditions of film preparation, the composite can form networks of different dimension even though the components are not changed, hence the position for the peak of GMR could move greatly. We also find all the calculated positions for largest GMR are lower than pc , which agrees with the recent observations [3]. Wang et al. [1] found the largest GMR occurred at p = 0.15 for Fe-based alloys and at p = 0.25 for Co-based alloys. We conjecture it may be due to the different pc in their samples. In Fig. 2, we studied the dependence of GMR on volume fraction for different conductivities of nonmagnetic components. The position for largest GMR turns to lower slightly with smaller σn , which fits experimental reports. In the GMR reports of Fe-based alloys [3], it showed the peak position became slightly lower when the nonmagnetic component changed from Ag to Cu. Our calculated results also show that a large σn is not beneficial to GMR.
Y. Ju, Z. Li / Physics Letters A 277 (2000) 169–174
Fig. 2. The volume-fraction dependence of GMR for different conductivity of nonmagnetic components in a two-dimensional network with pc = 0.5: (a) for σn = 2, (b) for σn = 4 and (c) for σn = 6. (N = σc+ /σc− = 10, N 0 = σs+ /σs− = 36, a = r − σc− , W1 = 1.4 and W2 = 0.6.)
Figs. 3 and 4 show the GMR dependence on volume fraction for different spin-asymmetric factors and different spin-polarizations of electrons. It is known that the discrepancy of conductivity between spin-up and spin-down channels gives rise to the GMR effect [5,8,9]. However, conductivity is related with the scattering strength and with the number of conduction electrons per unit volume [3]. Therefore, the discrepancy should include two aspects: one is the difference of scattering strength to electrons, which is denoted by spin-asymmetric factors, the other is the difference of electron number in each channel, which is denoted by the spin-polarizations of electrons. However, the latter is usually ignored by other researchers. Our results present that the greater spin-asymmetry factors and spin-polarization of electrons, the larger GMR effect will be. Figs. 5 and 6 show the GMR dependence on the particle sizes for a two-dimensional network film and for a three-dimensional network film, respectively. Our results are in agreement with former theoretical studies [5,6,8,9]. It shows that there is always an optimal particle size for GMR.
173
Fig. 3. The volume-fraction dependence of GMR for different spin-asymmetric factors in a two-dimensional network with pc = 0.5: (a) for N = 5, (b) for N = 10 and (c) for N = 15. (N 0 = σs+ /σs− = 36, a = r − σc− , σn = 2, W1 = 1.4 and W2 = 0.6.)
Fig. 4. The volume-fraction dependence of GMR for different spin-polarization of electrons in a two-dimensional network with pc = 0.5: (a) for W1 = 1.2, W2 = 0.8, (b) for W1 = 1.4, W2 = 0.6 and (c) for W1 = 1.6, W2 = 0.4. (N = σc+ /σc− = 10, N 0 = σs+ /σs− = 36, a = r − σc− , σn = 2.)
4. Summary Within the framework of effective cluster model, we have studied the GMR effect for magnetic alloys. The
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Y. Ju, Z. Li / Physics Letters A 277 (2000) 169–174
spin-dependent scattering occurring both within magnetic entities and at the interfaces between isolated magnetic particles and nonmagnetic matrix are taken into account. We have explained the GMR dependence on volume fraction. We find the position for the maximum of GMR depends greatly on the internal structures of films and is always lower than the percolation threshold of the composite. The magnitude of GMR is effected by some factors such as the discrepancy of electron scattering strength, the spin-polarization of electrons, the conductivity of the nonmagnetic matrix and the particle size.
Acknowledgement
Fig. 5. The particle-size dependence of GMR for a three-dimensional network with pc = 1/3. (N 0 = σs+ /σs− = 36, σn = 2, W1 = 1.4 and W2 = 0.6, p = 0.22.)
This subject was supported by the National Natural Science Foundation of China under Grant No. 19774042.
References
Fig. 6. The particle-size dependence of GMR for a two-dimensional network with pc = 0.5. (N 0 = σs+ /σs− = 36, σn = 2, W1 = 1.4 and W2 = 0.6, p = 0.4.)
[1] J.-Q. Wang, G. Xiao, Phys. Rev. B 49 (1994) 3982. [2] F. Parent, J. Tuaillon, L.B. Stern, V. Dupuis, B. Prevel, A. Perez, P. Melinon, G. Guiraud, R. Morel, A. Barthélémy, A. Fert, Phys. Rev. B 55 (1997) 3683. [3] A. Gerber, A. Minler, B. Groisman, M. Karpovsky, A. Gladkikh, A. Sulpice, Phys. Rev. B 55 (1997) 6446. [4] S. Rubin, M. Holdenried, H. Micklitz, J. Magn. Magn. Mater. 203 (1999) 97. [5] S. Zhang, P.M. Levy, J. Appl. Phys. 73 (1993) 5315. [6] R.Y. Gu, L. Sheng, D.Y. Xing, Z.D. Wang, J.M. Dong, Phys. Rev. B 53 (1996) 11685. [7] C.R. Chang, K.H. Lo, J. Appl. Phys. 80 (1996) 6888. [8] R. Yang, W. Zhang, W.J. Song, J. Appl. Phys. 84 (1998) 2044. [9] R. Yang, W.J. Song, J. Phys. Condens. Matter 11 (1999) 4341. [10] W.T. Doyle, J. Appl. Phys. 85 (1999) 2323. [11] B.L. Johnson, R.E. Camley, Phys. Rev. B 44 (1991) 9997. [12] X. Liu, Z.Y. Li, Phys. Lett. A 223 (1996) 475.
Physics Letters A 277 (2000) 169–174 www.elsevier.nl/locate/pla
The study on the giant magnetoresistance in magnetic granular alloys Yan Ju a , Zhenya Li a,b,∗ a Department of Physics, Suzhou University, Suzhou 215006, China b CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China
Received 2 August 2000; accepted 26 October 2000 Communicated by J. Flouquet
Abstract To study the giant magnetoresistance in magnetic granular alloys, an effective cluster model is extended to include spindependent interface scattering and bulk scattering. The dependence of giant magnetoresistance on particle size and volume fraction of magnetic component is discussed, some features of the giant magnetoresistance observed in recent experiments can be explained and the optimal choice of parameters to maximize the giant magnetoresistance can be determined. 2000 Elsevier Science B.V. All rights reserved. PACS: 75.70.Pa; 75.70.Ak; 75.70.-i; 75.50.Tt Keywords: Giant magnetoresistance; Granular composite; Effective cluster model
1. Introduction The giant magnetoresistance (GMR) in magnetic granular alloys has been studied thoroughly, since it provides a new perspective for technological applications. It is generally believed that GMR is mainly due to the spin-dependent interfacial scattering as well as the spin-dependent bulk scattering. However, there are still some aspect lacking good theoretical explanation, such as the volume fraction dependence of GMR in granular alloys. The relevant experimental reports [1–3] demonstrate some universal features: (1) The GMR reaches a maximum at the volume fraction of magnetic component p = 0.15–0.25 [1,2] or about 0.4 [3]. (2) As p further increases beyond the
* Corresponding author. Fax: +86-512-5112907.
E-mail address: [email protected] (Z. Li).
peak region, the GMR drops and GMR ' 0 at p → 1. These workers attributed it to the increasing coalescence of magnetic particles with the increment of p. The formed large multidomain clusters have no contribution to GMR [4], hence the largest GMR occurs below the percolation threshold of the system [3]. Recently, a few theoretical studies were developed on the GMR for the granular alloys. Some works [5,6] had respectively explained GMR vs. annealing temperature successfully, but had failed the GMR vs. volume fraction of magnetic component. Some works [7] made an attempt to include the mutual interactions, however, their model could not be applied to high volume fraction region, so the results was quite different from experimental observations. Within the framework of effective medium approximation (EMA), the peak of GMR was obtained [8,9], but there were still great discrepancy with the recent experimental data [1–3] even compared with their best results.
0375-9601/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 0 ) 0 0 7 0 9 - X
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Y. Ju, Z. Li / Physics Letters A 277 (2000) 169–174
EMA, which assumes the components distributed in highly random fashion, is a useful method to treat the transport properties of granular composites. Nevertheless, it seems to be quite unsatisfactory for investigation the effective properties of systems due to ignoring the realistic structures of the medium. Higher multipole interactions produce strong short range forces that drive similar particles clustering during sample preparation [10]. Based on the simple effective cluster model, Doyle [10] had got good theoretical results on the wavelength dependent optical transmission of granular CoSiO2 films over the entire range of metal volume loading. In this Letter, we extended the effective cluster model to research GMR in granular alloys, with the consideration of spin-dependent interface scattering and bulk scattering. Our results could well explain the observed dependence of GMR on size and volume fraction of the ferromagnetic component.
2. Formalism 2.1. The general expression for the conductivity in the effective cluster model We first consider the composite consists of two kinds of granules with conductivity σ1 and σ2 , present in volume fractions p and 1 − p, respectively. The granule 1 with σ1 exists both in the region of compact cluster and in the region of being isolated particle surrounded by granule 2 with σ2 . These two regions intermingle with each other and exactly fill the whole space. The volume fraction of the cluster region is V1 and the one of the uncluster region is V2 . Denoting the fraction of granule 1 in the cluster region by f , it is easy to know that V1 = fp
(1)
and V2 = 1 − fp.
(2)
The conductivity of cluster region σcluster and uncluster region σmg , treated as constituent conductivity, are inserted in EMA equation as (1 − fp)
σmg − σemt σcluster − σemt + fp = 0; (3) σmg + 2σemt σcluster + 2σemt
here, σemt is the effective conductivity of the composite which we search for. The fraction of granule 1 in the uncluster region is (1 − f )p. Therefore the actual volume fraction p0 of granule 1 in the uncluster region, relative to the volume of uncluster region, is p0 =
(1 − f )p (1 − f )p . = V1 1 − fp
(4)
Hence it is straightforward to derive the effective conductivity of the uncluster region by the Maxwell– Garnett equation σmg − σ2 (1 − f )p σ1 − σ2 = . σmg + 2σ2 1 − fp σ1 + 2σ2
(5)
2.2. The application to GMR in ferromagnetic alloys Let us consider a system composed ferromagnetic granules (such as Co, Ni, Fe) and nonmagnetic granules (such as Cu, Ag, Au), presenting in volume fraction p and 1 − p, respectively. The ferromagnetic granules exist in two regions: (1) cluster region with α , formed by compact ferromagnetic granules; σcluster α , being isolated single do(2) uncluster region with σmg main ferromagnetic particle surrounded by nonmagnetic granules. We define the superscript α = +(−) for the electron spin parallel (antiparallel) to the magnetic moment of ferromagnetic component. Coated with shells which are thin regions of mixed ferromagnetic and nonmagnetic atoms [11], the isolated particles could be regarded as equivalent solid particles [12] with σ˜ α : σ˜ α =
γ α (1 + 2γ α ) + 2λγ α (1 − γ α ) α σc ; (1 + 2γ α ) − λ(1 − γ α )
(6)
σα
a 3 ) . σcα and σsα are the here, γ α = σsα , λ = ( a+t c conductivities of core and shell of isolated particles, respectively. a is the radius of the core and t is the thickness of the shell. Taking the limit t → 0, the ratio γ α = t/σsα has a finite value [6,8,9] and σ˜ α is reduced to
σ˜ α =
a σ α. a + γ α σcα c
(7)
According to Eq. (5) we can get the effective conducα of the uncluster region, tivity σmg
Y. Ju, Z. Li / Physics Letters A 277 (2000) 169–174 α −σ σmg n α σmg
+ 2σn
=
(1 − f )p σ˜ α − σn , 1 − fp σ˜ α + 2σn
(8)
where σn is conductivity of nonmagnetic granule. The spins of electrons only have two directions, parallel and antiparallel to +z axis (such as the direc→ − tion of the applied magnetic field H ), hence electrons are divided into spin-up and spin-down channels correspondingly. The number of electrons in each chan− → → − nel is equal without H . Applied strong H , the two channels become uneven, for electrons skipping from spin-down channel to spin-up channel. It is known [2] that conductivity depends on not only the scattering strength but also the number of conduction electrons per unit volume. Therefore, at present of applied → − field H , the conductivity of each entities in spin-up and spin-down channels should be revised correspond− → ingly, relative to the one without H , by multiplying by W1 and W2 which indicate the change of electron number in spin-up and spin-down channels, i.e., the spin-polarization of electrons. It is nature that W1 > 1 and W2 < 1. Magnetoresistance (MR) is defined as the difference of resistivity between the completely magnetized state and the completely demagnetized stated. For the magnetized state, all the magnetic components have same magnetization direction parallel to the strong applied − → field H (point up), so the effective conductivity of the composite is
171
magnetization pointed up and pointed down, so the effective conductivity σemt in each channel should be equal. It can be derived as + − −σ σmg − σemt 1 − fp σmg emt + + − 2 σmg + 2σemt σmg + 2σemt + − − σemt σcluster fp σcluster − σemt + = 0. + + − 2 σcluster + 2σemt σcluster + 2σemt (12) Hence the effective conductivity of σ (0) ≡ σ (H = 0) of the composite should be σ (0) = 2σemt .
(13)
Then the GMR of the composite can be represented as follows: σ (H ) − σ (0) . GMR = (14) σ (H ) 2.3. Discussion about the cluster fraction f
= 0,
(10)
= 0.
(11)
The fraction f of ferromagnetic component in clusters plays a critical role in the effective cluster model. In general, the cluster fraction should be a function f (p) of the relative volume loading of magnetic component. The proper choice of f depends upon the preparation of the composites. The magnetic alloys [1–3] were made by cosputtering or coevaporation methods that deposited the components in atomic dispersion. If sputtered atoms and molecules arriving at the surface layer during sample preparation have a relatively high surface mobility, they will form two-dimensional network in each sputtered layer, even though the film is of three dimensions [10]. Hence, the cluster fraction f should be in the form of two dimensions and it should be related with percolation threshold pc of the composite for the reason that it is easy to form cluster if pc is relatively low. Therefore, we assume 1 for p < 12 + pc , [1−(1−(1/2−pc+p))1/2 ]2 f = [1−(1−(1/2−p (15) 1/2 ]2 +[1−(1/2−p +p)1/2 ]2 c +p)) c for p > 12 + pc .
→ − For the demagnetized state, no field H is applied. The spin-up and spin-down channels are even and there are the same numbers of magnetic components with
In fact, it turns to just the function f2 in Ref. [10], when taking a characteristic two-dimensional percolation threshold pc = 0.5. Doyle has achieved good agreement with experiments for optical transmission
+ − + σemt . σ (H ) = σemt + σemt
(9)
− σemt
and are the effective conductivity for Here, spin-up and spin-down channels. According to Eq. (3), we can get, respectively, (1 − fp) + fp
+ + (W1 σmg ) + 2σemt + + (W1 σcluster ) − σemt
+ + (W1 σcluster ) + 2σemt
(1 − fp) + fp
+ ) − σ+ (W1 σmg emt
− ) − σ− (W2 σmg emt
− − (W2 σmg ) + 2σemt − − (W2 σcluster ) − σemt
− − (W2 σcluster ) + 2σemt
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Y. Ju, Z. Li / Physics Letters A 277 (2000) 169–174
measurements and conductivity measurements of Au– SiO2 samples, by adopting the function f2 [10]. On the other hand, the sputtered atoms and molecules will form three-dimensional network, if they have a relatively low surface mobility. We assume, in a similar way, cluster fraction f in a three-dimensional form as 1 for p < 12 + pc , [1−(1−(1/2−pc +p))1/3 ]3 f = [1−(1−(1/2−p (16) 1/3 ]3 +[1−(1/2−p +p)1/3 ]3 c +p)) c for p > 12 + pc .
3. Discussion In this Letter, the GMR effect for magnetic alloys is studied according to Eqs. (7)–(16) and the influence of volume fraction of magnetic component and the particle size on GMR are discussed. For the reason that the compact multidomain cluster have no contribution − + = σcluster = 1 (arbitrary to GMR [4], we select σcluster units). It is known that the conduction electrons experience strong scattering and hence result in low conductivity, if their spins are antiparallel to the magnetic moments of the ferromagnetic particles. Therefore, we take a value lower than σcluster for the conductivity σc− such as 0.8 in our calculation. Figs. 1–4 study the dependence of GMR on the volume fraction p of ferromagnetic contents. There is a peak of GMR in our calculations. The ferromagnetic components exist in the form of compact clusters and isolated particles. It is only the isolated particles that contribute to GMR. In the magnetically dilute region, ferromagnetic components mainly exist in the form of isolated particles, so GMR increases with p due to the increasing concentration of magnetic scattering centers. As p further increases, cluster fraction raises rapidly and GMR drops beyond the peak region. When p → 1, most of magnetic components coalesce into clusters, hence there are no GMR effect. We find that the position for the maximum of GMR mainly depends on the internal structures of films. In Fig. 1, we present GMR vs. volume fraction for a two-dimensional network with pc = 0.5 and for three-dimensional networks with pc = 1/3 and 0.2, respectively. The calculation results show that the largest GMR occurs at about p = 0.42, 0.26 and 0.18.
Fig. 1. The volume-fraction dependence of GMR: (a) for two-dimensional network with pc = 0.5, taking W1 = 1.4 and W2 = 0.6; (b) and (c) for three-dimensional networks with pc = 1/3 and pc = 0.2, taking W1 = 1.8 and W2 = 0.2.
Wang et al. [1] and Gerber et al. [3] studied GMR for CoAg alloys, respectively, and found the peak of GMR at about p = 0.25 and 0.40–0.42. Our results are in good agreement with their experimental reports. With different conditions of film preparation, the composite can form networks of different dimension even though the components are not changed, hence the position for the peak of GMR could move greatly. We also find all the calculated positions for largest GMR are lower than pc , which agrees with the recent observations [3]. Wang et al. [1] found the largest GMR occurred at p = 0.15 for Fe-based alloys and at p = 0.25 for Co-based alloys. We conjecture it may be due to the different pc in their samples. In Fig. 2, we studied the dependence of GMR on volume fraction for different conductivities of nonmagnetic components. The position for largest GMR turns to lower slightly with smaller σn , which fits experimental reports. In the GMR reports of Fe-based alloys [3], it showed the peak position became slightly lower when the nonmagnetic component changed from Ag to Cu. Our calculated results also show that a large σn is not beneficial to GMR.
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Fig. 2. The volume-fraction dependence of GMR for different conductivity of nonmagnetic components in a two-dimensional network with pc = 0.5: (a) for σn = 2, (b) for σn = 4 and (c) for σn = 6. (N = σc+ /σc− = 10, N 0 = σs+ /σs− = 36, a = r − σc− , W1 = 1.4 and W2 = 0.6.)
Figs. 3 and 4 show the GMR dependence on volume fraction for different spin-asymmetric factors and different spin-polarizations of electrons. It is known that the discrepancy of conductivity between spin-up and spin-down channels gives rise to the GMR effect [5,8,9]. However, conductivity is related with the scattering strength and with the number of conduction electrons per unit volume [3]. Therefore, the discrepancy should include two aspects: one is the difference of scattering strength to electrons, which is denoted by spin-asymmetric factors, the other is the difference of electron number in each channel, which is denoted by the spin-polarizations of electrons. However, the latter is usually ignored by other researchers. Our results present that the greater spin-asymmetry factors and spin-polarization of electrons, the larger GMR effect will be. Figs. 5 and 6 show the GMR dependence on the particle sizes for a two-dimensional network film and for a three-dimensional network film, respectively. Our results are in agreement with former theoretical studies [5,6,8,9]. It shows that there is always an optimal particle size for GMR.
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Fig. 3. The volume-fraction dependence of GMR for different spin-asymmetric factors in a two-dimensional network with pc = 0.5: (a) for N = 5, (b) for N = 10 and (c) for N = 15. (N 0 = σs+ /σs− = 36, a = r − σc− , σn = 2, W1 = 1.4 and W2 = 0.6.)
Fig. 4. The volume-fraction dependence of GMR for different spin-polarization of electrons in a two-dimensional network with pc = 0.5: (a) for W1 = 1.2, W2 = 0.8, (b) for W1 = 1.4, W2 = 0.6 and (c) for W1 = 1.6, W2 = 0.4. (N = σc+ /σc− = 10, N 0 = σs+ /σs− = 36, a = r − σc− , σn = 2.)
4. Summary Within the framework of effective cluster model, we have studied the GMR effect for magnetic alloys. The
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spin-dependent scattering occurring both within magnetic entities and at the interfaces between isolated magnetic particles and nonmagnetic matrix are taken into account. We have explained the GMR dependence on volume fraction. We find the position for the maximum of GMR depends greatly on the internal structures of films and is always lower than the percolation threshold of the composite. The magnitude of GMR is effected by some factors such as the discrepancy of electron scattering strength, the spin-polarization of electrons, the conductivity of the nonmagnetic matrix and the particle size.
Acknowledgement
Fig. 5. The particle-size dependence of GMR for a three-dimensional network with pc = 1/3. (N 0 = σs+ /σs− = 36, σn = 2, W1 = 1.4 and W2 = 0.6, p = 0.22.)
This subject was supported by the National Natural Science Foundation of China under Grant No. 19774042.
References
Fig. 6. The particle-size dependence of GMR for a two-dimensional network with pc = 0.5. (N 0 = σs+ /σs− = 36, σn = 2, W1 = 1.4 and W2 = 0.6, p = 0.4.)
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