Grain Size Effect on Electrical Conductivity and ... - Chin. Phys. Lett.

CHIN. PHYS. LETT. Vol. 26, No. 11 (2009) 117502

Grain Size Effect on Electrical Conductivity and Giant Magnetoresistance of Bulk Magnetic Polycrystals * LUO Wei(罗威), ZHU Lin-Li(朱林利), ZHENG Xiao-Jing(郑晓静)** Key Laboratory of Mechanics on Western Disaster and Environment (Ministry of Education) and Department of Mechanics and Engineering Science, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000

(Received 15 July 2009) By solving the Boltzmann transport equation and considering the spin-dependent grain boundary scattering, the distribution of electrons in grains and the electrical transport properties in the applied magnetic field are studied. With regard to the dominant influence of grain boundary scattering which is taken as a boundary condition for the electrical transport, the grain size-dependent electrical conductivity is investigated. In addition, the reorientation of the relative magnetization between grains brings the change of the electron spin when the magnetonanocrystalline material is subjected to the magnetic field, resulting in the remarkable giant magnetoresistance effect.

PACS: 75. 47. De, 73. 43. Qt, 72. 20. −i The size effect of electrical conductivity originates from electrical conductivity decreasing with film thickness and wire diameter, and the Fuchs– Sondheimer[1−2] (FS) and the Mayadas–Shatzkes (MS)[3] models have been established by considering the surface effect and the grain boundary effect. As the dimension of structure continues to diminish, size effect of low-dimensional materials becomes more prominent and the electrical conductivity is much smaller than the corresponding bulk value.[4−7] It is important to investigate this effect due to the abroad application of the bulk polycrystalline materials in the big thermal load.[8] The experimental researches[9−10] indicate that bulk electrical conductivity reduces along with the decreasing grain size. However, size effect of bulk-material electrical conductivity, which is different from that of films and wires, is attributed to the reduction of the interior mirostructure and grain; and the MS model describing the grain boundary effect of thin films can be applied to qualitatively explain the size effect of bulk materials but is non-strict and full of arguments.[11−13] By studying the transport properties of electrons in grains, the grain size effect of electrical conductivity could be analyzed, while the previous investigations have scarcely discussed it in the sight of grains. In past decades, numerous studies in experiment[14−20] and in theory[21−24] have been focusing on the magnetic multilayer composites, heterogeneous superlattice thin films and Mn, Gd, Sr cluster-alloy compounds which display the so-called giant magnetoresistance effect. However, there are few investigations on discussing the electric properties of magnetic nanocrystalline materials under the

magnetic field, especially the effect of the grain size on the magnetoresistance theoretically. In this Letter, a model is proposed to describe the grain size effect on the electrical conductivity of magnetic nanocrystalline materials by considering the electron spin and the corresponding transport in the microstructures. The Boltzmann equation for this problem is solved to find electrical conductivity considering the grainboundary scattering as the boundary conditions in grains, which can not only account for the various grain shapes conveniently in detail but also avoid the invalidation of Matthiessen’s rule. Moreover, the basic assumption underlying the description is that there exists the antiferromagntic coupling among grains, because of the antiferromagnetic secondary phases in grain boundaries[25] or magnetization through a metamagnetic transition from an antiferromagnetic state to a ferromagnetic state,[26] and thus the magnetization intensity between two grains transform the relative direction from the antiparallel one to the parallel in the applied magnetic field, leading to the change of the electron spin and the spin-dependent grain-boundary scattering. The spin asymmetry results from the corresponding asymmetry in the density of the states for electrons of different spin. Motivated from this point, we will investigate the dependence of the electrical conductivity on the applied magnetic field (magnetoresistance effect). When the external electric field 𝐸 is in the direction of the 𝑥 coordinate, the electron distribution function of the periodic unit can be considered as a function of the coordinate 𝑥 due to the grain-boundary scattering. Therefore, the Boltzmann equation reduces to a differential equation depending on the coordinate 𝑥 in grains,

* Supported by the National Natural Science Foundation of China under Grant No 90405005, the National Basic Research Program of China under Grant No 2007CB607506, the Specialized Research Fund for the Doctoral Programme of Higher Education of China under Grant No 20050730016, and the Fund of of Lanzhou University under Grant No WUT2005Z04. ** Email: [email protected] c 2009 Chinese Physical Society and IOP Publishing Ltd ○

117502-1


𝑔 𝜕𝑔 𝑒𝐸𝑥 𝜕𝑓0 + = , 𝜕𝑥 𝜏↑(↓) 𝑣𝑥 𝑚𝑣𝑥 𝜕𝑣𝑥

(1)

𝑔+↓ (0, 𝑣𝑥 ) = 𝑃↓↓ 𝑇↓ 𝑔𝑃 +↓ (𝑑, 𝑣𝑥 ) + 𝑃↓↑ 𝑇↑ 𝑔𝑃 +↑ (𝑑, 𝑣𝑥 )

where 𝑓0 is the equilibrium distribution function and 𝑔 is the correction to the distribution function; 𝜏↑(↓) is a relaxation time to describe the scattering from other sources (point defects and phonons), in which ↑ and ↓ denotes the electron up-spin and down-spin, respectively; 𝑒 is the electron charge, and 𝑚 and 𝑣𝑥 are the electron mass and 𝑥 component of the electron velocity, respectively. Thus, the electron distribution function in grains can be written as 𝑓 = 𝑓0 + 𝑔(𝑣, 𝑥). Note that the terms from departure of Ohm law and the magnetic field in Boltzmann equation are negligibly small.[2,21] One can easily obtain the solution of the distribution function from Eq. (1) as follows: ⎧ 𝑒𝐸𝜏↑(↓) 𝜕𝑓 {︁ 0 ⎪ ⎪ 𝑚 𝜕𝑣𝑥 1 + 𝐹+↑(↓) ⎪ ⎪ [︁ ]︁}︁ ⎪ ⎪ ⎪ ⎨ · exp 𝜏 ∓𝑥|𝑣𝑥 | , 𝑣𝑥 > 0, ↑(↓) {︁ (2) 𝑔(𝑣𝑥 , 𝑥) = 𝑒𝐸𝜏↑(↓) 𝜕𝑓0 ⎪ ⎪ 1 + 𝐹 ⎪ −↑(↓) 𝑚 𝜕𝑣𝑥 ⎪ ⎪ ⎪ [︁ ]︁}︁ ⎪ ⎩ ∓𝑥 · exp 𝜏↑(↓) , 𝑣𝑥 < 0, |𝑣𝑥 |

𝑔−↑ (𝑑, 𝑣𝑥 ) = 𝑃↑↑ 𝑇↑ 𝑔𝑁 −↑ (0, 𝑣𝑥 ) + 𝑃↑↓ 𝑇↓ 𝑔𝑁 −↓ (0, 𝑣𝑥 )

+ 𝑅↓ 𝑔𝑄−↓ (0, 𝑣𝑥 ),

where 𝐹±↑(↓) is an arbitrary function of the velocity, which can be determined from the boundary condition. α

P

y

d

Q

+ 𝑅↑ 𝑔𝑄+↑ (𝑑, 𝑣𝑥 ), 𝑔−↓ (𝑑, 𝑣𝑥 ) = 𝑃↓↓ 𝑇↓ 𝑔𝑁 −↓ (0, 𝑣𝑥 ) + 𝑃↓↑ 𝑇↑ 𝑔𝑁 +↑ (0, 𝑣𝑥 ) + 𝑅↓ 𝑔𝑄+↓ (𝑑, 𝑣𝑥 ),

where 𝑑 is the size of the grain in polycrystalline materials, 𝑃 represents the preceding unit, and 𝑁 represents the nest unit in the 𝑥 direction; and there are two distribution functions, 𝑔+ for electrons with 𝑣𝑥 > 0 and 𝑔− for electrons with 𝑣𝑥 < 0. Because of the assumption of the perfect grain units with periodic boundaries in polycrystalline materials, the electron distribution function is consistent in each unit, resulting in 𝑔−↑(↓) = 𝑔𝑅−↑(↓) , 𝑔+↑(↓) = 𝑔𝑃 +↑(↓) . Having found the various 𝑔’s, and introducing polar coordinate (𝑣, 𝜃, 𝜙) in the 𝑣-space (𝑣𝑧 = 𝑣 cos 𝜃), we can obtain the current density in the grain, ∫︁ ∫︁ {︂(︂ ∫︁ 𝜋2 ∫︁ 2𝜋 )︂ 𝑒2 𝑚2 𝐸𝑥 ∞ 𝜋 𝐽𝑥 (𝑥) = − 𝑑𝜙 + 𝑑𝜙 3𝜋 ℎ3 0 0 0 2 )︂)︂ [︂ (︂ (︂ −𝑥 · cos2 𝜙 𝜏↑ 1 + 𝐹+↑ exp 𝜆↑ sin 𝜃 cos 𝜙 )︂)︂]︂ (︂ (︂ −𝑥 + 𝜏↓ 1 + 𝐹+↓ exp 𝜆↓ sin 𝜃 cos 𝜙 [︂ (︂ ∫︁ 3𝜋 2 + 𝑑𝜙 cos2 𝜙 𝜏↑ 1 + 𝐹−↑ 𝜋 2

R

)︂)︂ (︂ −𝑥 · exp 𝜆↑ sin 𝜃 cos 𝜙 )︂)︂]︂}︂ (︂ (︂ −𝑥 + 𝜏↓ 1 + 𝐹−↓ exp 𝜆↓ sin 𝜃 cos 𝜙 𝜕𝑓0 · 𝑣3 sin3 𝜃𝑑𝜃𝑑𝑣, (4) 𝜕𝑣

β x

E

Fig. 1. Schematic diagram of microstruture of bulk polycrystals.

For the boundary conditions in grains, the parameters 𝑇(↑↓) and 𝑅(↑↓) for the grain boundary effect is introduced as the transmisitivity and reflectivity of the conduction electrons with tunnelling the grain boundary barrier from a grain to the next one.[22] It is assumed that the direction of spin for the electrons is changed when electrons tunnel the grain boundary in the applied magnetic field. Another parameter introduced here is 𝑃𝑖𝑗 , which denotes the probability of electrons from 𝑗 state of spin to 𝑖 state of spin after the transmission between two grains, and we have the corresponding correlations such as 𝑃↑↑ = 𝑃↓↓ = cos2 (𝜗/2) and 𝑃↓↑ = 𝑃↑↓ = sin2 (𝜗/2), where 𝜗 is the angle of the magnetization intensity for the neighboring grains. As shown in Fig. 1, based on the electrons balance at the boundaries 𝛼 and 𝛽 between two grains, the boundary conditions in the periodic unit can be written as 𝑔+↑ (0, 𝑣𝑥 ) = 𝑃↑↑ 𝑇↑ 𝑔𝑃 +↑ (𝑑, 𝑣𝑥 ) + 𝑃↑↓ 𝑇↓ 𝑔𝑃 +↓ (𝑑, 𝑣𝑥 ) + 𝑅↑ 𝑔𝑄−↑ (0, 𝑣𝑥 ),

(3)

where 𝜆↑↓ = 𝜏↑↓ 𝑣, 𝑣 = |𝑣| is the background free path of the electrons. Consequently, the effective electrical conductivity can be defined as ∫︁ 𝑑 1 𝜎= 𝐽𝑥 (𝑥)𝑑𝑥. (5) 𝐸𝑥 𝑑 0 In considering the hard axis magnetization and the 𝐻 first-order approximation,[27] namely, cos 𝜗2 = 𝐻𝑎 = 𝑀 , we obtain 𝑀𝑠 (︁ 𝑀 )︁2 𝑃↑↑ = 𝑃↓↓ = cos2 (𝜗/2) = , 𝑀𝑠 (︁ 𝑀 )︁2 𝑃↓↑ = 𝑃↑↓ = sin2 (𝜗/2) = 1 − , (6) 𝑀𝑠 where 𝑀𝑠 is the saturation magnetization. The relative change of the resistivity (magnetoresistance) is defined as 𝜌↑↓ − 𝜌↑↑ ⃒⃒ 𝜎 ↑↑ − 𝜎 ↑↓ ⃒⃒ 𝜒= = , (7) ⃒ ⃒ 𝜌↑↑ 𝜎 ↑↓ Ψ=𝜋 Ψ=𝜋

117502-2


where 𝜌↑↓ and 𝜌↑↑ are the resistivities with antiparallel and parallel alignments of the intercrystalline magnetizations, respectively, and 𝜌𝜉𝜂 = (𝜎 𝜉𝜂 )−1 . From the expression of the electrical conductivity in Eqs. (4)–(7), one can note that the resistivity of nanocrystalline materials is dependent on the physical parameters, e.g., the electron mean free path, the size of the microstructures such as grain size, and the applied magnetic field.

of scattering is much higher. Because the spin asymmetry of scattering is the description of the roughness of grain boundary, it is apparent that the roughness of grain boundary greatly affects the electrical conductivity and the relative change of the resistivity.

Fig. 2. 𝑇↑ = 0.6, 𝜆 = 80 nm, 𝑃↑↑ = 0.5. The electrical conductivity as a function of the grain size at different 𝑁𝑠 .

Fig. 3. 𝑇↑ = 0.6, 𝜆 = 80 nm. The relative change of the resistivity as a function of the grain size at different 𝑁𝑠 .

Fig. 4. Relative change of the resistivity varying with the applied magnetic field (a) for different electron mean free paths, 𝑑 = 10 nm, 𝑇↑ = 0.5, 𝑁𝑠 = 10, (b) for different spin asymmetries of scattering, 𝑑 = 5 nm, 𝑇↑ = 0.5, 𝜆 = 80 nm, and (c) for different grain sizes, 𝑇↑ = 0.6, 𝑁𝑠 = 10.0, 𝜆 = 80 nm.

The electrical conductivity as a function of the grain size is presented in Fig. 2 with different 𝑁𝑠 , which describes the spin asymmetry of the scattering of electrons at grain boundaries and is defined as 𝑁𝑠 = 𝐷↑ /𝐷↓ , where 𝐷↑(↓) = 1 − 𝑇↑(↓) . The electrical conductivity decreases acutely when the grain size is less than 100 nm. For the same grain size, the electrical conductivity increases with the augmentation of the asymmetry of spin scattering. In Fig. 3, the grain-size dependent magnetoresistance is shown with the different spin asymmetries of scattering. It can be found that the magnetoresistance rises with the increasing grain size. The giant magnetoresistance (GMR) effect becomes more distinct when the grain size is less than 20 nm and the spin asymmetry

Furthermore, we investigate the magnetoresistance varying with the parameters, e.g. the electron mean free path, the spin asymmetry of scattering, the grain size, in the applied magnetic field, respectively. The relative change of the resistivity vs magnetic field with different electron mean free paths is shown in Fig. 4(a). It can be found that the magnetoresistance decreases with the decreasing electron mean free path. The influence of the spin asymmetry of scattering on the resistivity is also analyzed as shown in Fig. 4(b). One can notice that the relative change of the resistivity increases with increasing 𝑁𝑠 . Figure 4(c) shows the relative change of the resistivity varying with the grain size. It can be noted that the relative change of the resistivity increases sharply as the grain size decreases,

117502-3


indicating the significant size-dependent magnetoresistance of nanocrystalline materials. We can also find from Fig. 4 that the relative change of the resistivity descreases with the increasing magnitude of the applied magnetic field. This is a matter of fact that the magnetization in the grain is in the easy axial for the vanished magnetic field and the direction of the magnetization of the neighboring grains is antiparallel. When acting on the applied magnetic field, the relative direction of the magnetization is changed from the antiparallel to the parallel, resulting in the resistivity decreasing. In conclusion, we have proposed a simple theoretical model to describe the grain size effect of polycrystalline materials by considering the spin of electrons and allowing for the influence of applied magnetic field on the transport properties of electrons, and discuss the GMR effect in bulk magnetic polycrystalline materials. It is found that the electrical conductivity and the magnetoresistance of bulk magnetic polycrystalline materials are affected evidently by the grain size. The present results will be helpful when studying the properties of magnetic nanocrystalline materials and useful in the engineering applications.

References

[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

[1] Fuchs K 1938 Proc. Camb. Phil. Soc. 34 100

117502-4

Sondheimer E H 1952 Adv. Phys. 1 1 Mayadas A F, Shatzkes M 1970 Phys. Rev. B 1 1382 Wu W et al 2004 Appl. Phys. Lett. 84 2838 K¨ astle G et al 2004 Phys. Rev. B 70 165414 Meyerovich A E and Ponnmarev I V 2002 Phys. Rev. B 65 155413 Meyerovich A E and Ponnmarev I V 2003 Phys. Rev. B 67 026302 Disalvo F J 1999 Science 285 703 Zhai P C et al 2006 Appl. Phys. lett. 89 052111 Mi J L, Zhu T J, Zhao X B and Tu J P 2007 J. Appl. Phys. 101 054314 Bietsch A and Michel B 2002 Appl. Phys. Lett. 80 3346 Landauer R 1957 IBM J. Res. Dev. 1 223 Vancea J, Reiss G and Hoffmann H 1987 Phys. Rev. B 35 6435 Gordeev S N et al 2001 Phys. Rev. Lett. 87 186808 Lebon A et al 2004 Phys. Rev. Lett. 92 037202 Wu J et al 2005 Phys. Rev. Lett. 94 037201 Yu S Y et al 2006 Appl. Phys. Lett. 89 162503 Koyama K et al 2006 Appl. Phys. Lett. 89 182510 Miao J H et al 2007 J. Appl. Phys. 101 043904 Sengupta K, Iyer K K and Sampathrumaran E V 2005 Phys. Rev. B 72 054422 Hood R Q, Falicov L M and Penn D R 1994 Phys. Rev. B 49 368 Camley R E and Barna´s J 1989 Phys. Rev. Lett. 63 664 Levy P M, Zhang S and Fert A 1990 Phys. Rev. Lett. 65 1643 Zhang S, Levy P M and Fert A 1992 Phys. Rev. B 45 8689 Chen P et al 2001 Phys. Rev. Lett. 87 107202 Zhang Y Q, Zhang Z D and Aarts J 2004 Phys. Rev. B 70 132407 O’Handley R C 2000 Modern Magnetic Materials Principles and Applications (New York: John Wiley & Sons, Ioc.)