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A study on magnetoresistivity of polycrystalline perovskite manganite films
8 October 2001
Physics Letters A 289 (2001) 106–110 www.elsevier.com/locate/pla
A study on magnetoresistivity of polycrystalline perovskite manganite films Yan Ju b , Zhen-Ya Li a,b,∗ a CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China b Department of Physics, Suzhou University, Suzhou 215006, China 1
Received 2 June 2001; accepted 28 August 2001 Communicated by J. Flouquet
Abstract Within the framework of effective medium theory and two-channel model, magnetoresistivity (MR) of polycrystalline perovskite manganite films are investigated. Grain boundaries are considered to result in the MR at low temperatures. Some features of recent experiments are explained qualitatively and the optimal choice of parameters to maximize MR in whole temperature region are discussed. 2001 Elsevier Science B.V. All rights reserved. PACS: 75.70.Pa; 75.70.Ak
1. Introduction The giant magnetoresistance effect (GMR) in magnetic granular alloys, considered to origin from asymmetrical scattering between spin-up and spin-down channels, has been studied thoroughly and has been found hard to meet application requirements well, such as huge resistivity change in the condition of low magnetic field B and high temperature T . Hence much attention was turned to other materials of intrinsically high degree of spin polarization, for example, the long researched doped perovskite manganite films. Recently, experiments [1] were made comparatively on epitaxial and polycrystalline perovskite manganite films (La0.67Sr0.33 MnO3 and La0.67Ca0.33 MnO3 ), * Corresponding author.
E-mail address: [email protected] (Z.-Y. Li). 1 Mailing address in China.
which demonstrated two features: (1) both of them exhibited a colossal magnetoresistivity effect (CMR) near critical temperature Tc , (2) contrasting to epitaxial films without MR at low temperatures (T Tc ), the MR in polycrystalline films increased almost linearly down to the lowest temperature. Offering significant MR over a wide temperature region, the polycrystalline perovskite manganite films were more attractive for applications, however theoretical researches on them were not enough. Though the CMR effect had been interpreted in some works [2], the mechanism of the MR in low temperature region was still confused. It was usually owned to the contribution of grain boundaries. But there was discrepancy in literature on the role of grain boundaries. It was proposed in Ref. [1] that spin-dependent scattering at grain boundaries, rather than magnetotunneling proposed by others [3], is responsible for the MR in low T region. In the case of qualities of carriers [4,5] and electronic structure [6], the doped perovskite manganite is
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 5 7 3 - 4
Y. Ju, Z.-Y. Li / Physics Letters A 289 (2001) 106–110
indeed a complicated system. However, its MR phenomenon has been studied by a model [2] extended from the treatment of magnetic granular alloys, which is based on such facts as: the transport in both systems is spin-dependent. Meanwhile, the low temperature behavior of MR in polycrystalline perovskite manganite films is much similar to the one in magnetic granular alloys [7,8]. It is nature to assume they due to the same mechanism of scattering on tiny ferromagnetic clusters as mentioned in Refs. [7,8]. Actually, tiny ferromagnetic Mn cluster may exist in perovskite manganite films due to local deviations from ferromagnetic order [9]. In this Letter, we present a model to study the polycrystalline perovskite manganite system, assuming the CMR effect be due to the near Tc mechanism inside grain regions, and the MR at low temperatures be due to the scattering by tiny Mn clusters existing in grain boundary regions, and experimental observations are interpreted qualitatively.
2. Formalism The transmission electron microscopy (TEM) of polycrystalline perovskite manganite films indicated that a well-defined grain morphology reminiscent of underlying polycrystalline SrTiO3 substrate, and furthermore, the wide-field Kerr microscope showed that the magnetic domains were defined by the grains [1]. Hence, we suppose that tiny ferromagnetic Mn clusters existing in grain regions of size of several µm are of strong coupling, such forming magnetic domains. However, in grain boundary regions, tiny ferromagnetic clusters are floppier than the bulk clusters [10] and of no interactions. Following spin-dependent twochannels model, MR is due to the asymmetric scattering in spin-up (+) and -down (−) channels, in which spins of carriers parallel and anti-parallel to the mag respectively. netic field B, Inside grain regions, when the spin of carrier is parallel to the magnetic moment (µ) of tiny Mn cluster, the carrier experiences weak scattering and hence a low resistivity (ρL ), otherwise, it experiences strong scattering and hence a high resistivity (ρH ). Taking a general spherical angle (θ, ϕ) between µ and B into account, the change of quantization axis from the magnetic moment direction to the magnetic field direction arises from a coordinate transformation characterized
by the corresponding rotation matrix cos(θ/2) sin(θ/2)e−iϕ . C= − sin(θ/2)eiϕ cos(θ/2)
107
(1)
In the simplified spin-dependent two-channel model, the two-point resistivity ρ(r , r ), which is forth-rank tensor, did not be taken into account, so that the resistivity ρ is a tensor of rank 2 [11] (see Ref. [11] for details). The bispinor ρL 0 ρ= 0 ρH transforms according to ρ = C−1 ρC, and after simple algebraic operations we got the usual form for resistiv± of grains as ity ρG ± = ρG
1 1 1 + cos(θ ) ± 1 − cos(θ ) 2 2
(2)
which is the diagonal terms of ρ . The nondiagonal terms of ρ average out with the azimuthal integration. ± Then taking the thermal average of ρG , 1 1 ± ρG (B, T ) = (ρH + ρL ) ∓ (ρH − ρL )L(α), 2 2
(3)
hence the conductivity σG± is σG± (B, T ) =
1
, ± ρG (B, T )
(4)
where ρH and ρL denotes the asymmetry of two channels. Note that the magnetoresistivity depends on the imbalance between the two conducting channels. If two channels are symmetrical, that is, ρH = ρL , no magnetoresistivity to be expected. The larger ρH /ρL , the larger magnetoresistivity is. In the following calculation, we take ρL = 1, ρH = 104 (arbit. unit) as Ref. [2]. L(α), equals to cos(θ ) , is Langevin function, with α = µBeff /KB T , and the effective magnetic field Beff includes the magnetic inter-cluster coupling described by a molecular field in addition to the external magnetic field [2]. The moment of the ferromagnetic Mn cluster depends on both the temperature and effective magnetic field, then we get 3Tc mG µB + , mG (B, T ) = L (5) Kb T T where mG is the reduced magnetization of grain regions.
108
Y. Ju, Z.-Y. Li / Physics Letters A 289 (2001) 106–110
Similarly, the resistivity ρB± of grain boundary regions, due to spin-dependent scattering on Mn cluster without coupling existing in the region of grain boundary, is treated quantum-mechanically and in the form of [7,8] J SZ J2 ± . ρB (B, T ) = ρ0 1 + 2 ∓ 2ρ0 (6) V V S Hence the conductivity σB± is σ ± (B, T )B =
1 , ρB± (B, T )
(7)
with reduced magnetization mB of grain boundary regions, (2S + 1)EH
SZ 2S + 1 = coth mB (B, T ) = S 2S kB T EH 1 coth − (8) . 2S kB T Here, V is the spin-independent potential and J is s–d exchange interaction, supposed to be smaller than V . ρ0 is the resistivity due to the scattering by potential V only. EH = gµB B/2 and g is the g factor of magnetic scatter taking value 2. In perovskite manganite films, carrier density in spin-up and -down channels is uneven, therefore the conductivities σG± and σB± must be revised as ± 2NG , (ρH + ρL ) ∓ (ρH − ρL )L(α) NB± , σB± (B, T ) = ρ0 (1 + J 2 /V 2 ) ∓ 2ρ0 (J /V )( SZ /S) (9)
σG± (B, T ) =
where N ± , the carrier density in each channel, is related with magnetic field and temperature and can be got easily from the expression of spin polarization of carrier [12] √ √ 3 1+m− 3 1−m N+ − N− =√ . P= + √ 3 N + N− 1+m+ 3 1−m It is obvious that N + = N − , in the case of P = 0, and ± NG (10) (B, T ) = 3 1 ± mG (B, T ) and NB± (B, T ) =
3
1 ± mB (B, T ).
(11)
Considering the polycrystalline films of island topology, i.e., region of grains embedded in a different host medium, namely the region of grain boundaries, ± of each channel is given the effective conductivity σeff in the form of Maxwell–Garnett equation [13] ± − σB± σeff
± σeff + 2σB±
= fG
σG± − σB±
σG± + 2σB±
,
(12)
where fG is the volume fraction of grain regions. If fG = 1, it returns to the case of epitaxial films. The total conductivity of σeff (B, T ) of the system is + − σeff (B, T ) = σeff (B, T ) + σeff (B, T ).
(13)
Then the MR of polycrystalline perovskite manganite film can be represented as follows: MR(T ) =
σeff (B, T ) − σeff (B = 0, T ) . σeff (B = 0, T )
(14)
3. Discussion and conclusion According to Eqs. (5) and (8)–(14), we derive the MR of polycrystalline perovskite manganite films over a wide T region and compare the calculated results with the experimental reports in Ref. [1]. In our calculation, we choose the magnetic moment of tiny cluster in grain regions µ = 20µB according to former theoretic studies [2], because the ferromagnetic clusters are found very small, and likewise the value of the total spin of cluster in grain boundary regions S = 13.5. Meanwhile, the value of ρ0 is taken larger than resistivity of grain region, such as 104 (arbit. unit), due to inhomogeneities of grain boundaries. The volume fraction of grain regions is selected as 0.9 and the magnetic field B as 4 T. In Fig. 1, the calculated MR in the case of polycrystalline films presents two components: (1) a steady incline of MR with increment of temperature in low T region, and (2) a sharp peak of MR near Tc , namely CMR effect. For comparison, we also deduce the case of epitaxial films by replacing fG = 1, that is, no grain boundary regions existing any more. There is only CMR effect left the near critical temperature where spin-fluctuation of ferromagnetic clusters is the largest, which is in agreement with former theoretic studies [2]. These results suggest that the CMR effect in doped perovskite manganite system be mainly
Y. Ju, Z.-Y. Li / Physics Letters A 289 (2001) 106–110
Fig. 1. The MR dependence on temperature for perovskite manganite films. (—) Epitaxial films, (– – –) polycrystalline films with the consideration of spin-polarization P , (– · – · –) polycrystalline films without the consideration of P .
caused by the spin-fluctuation of tiny clusters in grain regions. However, the MR in low T region is mainly caused by spin-dependent scattering on tiny clusters existing in grain boundary regions which are of no coupling and too small to be aligned by external magnetic field, hence the MR is enhanced continually till lowest temperature. In addition, the case of Tc = 230 K and Tc = 350 K are discussed, respectively, and find that, consistent with experimental reports [1] qualitatively, the CMR effect lessens with the increment of Tc . It maybe demonstrates why the shells of magnetic particles, according to the grain boundary regions in our model, could make a major contribution to GMR in magnetic granular alloys. It is an important presumption concluded from experiments and often cited in theoretic calculations. Because the Tc of magnetic metal is much larger than the above one, and it leads to a negligible CMR effect, but the spindependent scattering occurring at shells still results in large MR effect in low operating temperature. Note magnetoresistivity is an extra resistivity origining from the asymmetry of spin-down and -up channels which exhibits in two aspects. One is the imbalance of carrier density in both channels, indicated by the degree of spin polarization P of carriers. In Fig. 1, we have compared the cases with and without consideration of P . A larger MR will be got, if considered P . The other is connected with the degree of alignment of magnetic moments of ferromagnetic clusters (or particles). Both factors depends on temperature T and
109
Fig. 2. The MR dependence on temperature for different magnetic fields B.
Fig. 3. The MR and resistivity dependence on temperature for different spin-independent resistivity ρ0 .
It is shown in Fig. 2 that a stronger field B lead field B. to a sharper asymmetric broken in the system, hence a more manifest MR. Fig. 3 shows the MR and resistivity dependence on temperature for different spin-independent resistivity ρ0 of grain boundary region as 104 , 4 × 104 and 8 × 104 (arbit. unit), respectively. We find that grain boundary regions increase total resistivity of films and lead to the MR in low T region, but the CMR effect decreases simultaneously with the increment of ρ0 . Therefore, a high resistivity of films are not according to large MR in whole region of temperature. By the way, it is necessary to note that the calculated resistivity results cannot explain some experimental reports well, such as a sharp resistivity peak near Tc and the peak driven to high temperature with incre-
110
Y. Ju, Z.-Y. Li / Physics Letters A 289 (2001) 106–110
ment of magnetic field. It is possibly due to ignoring the qualities of carrier in our model, and more detail information on temperature and field dependence of ρH and ρL are needed which is the goal of our later work. On the other hand, in our model, it is assumed that the tiny ferromagnetic Mn clusters exist in the perovskite manganite films. However, the Mn cluster sizeeffect is neglected. In fact, the size of magnetic particles (Mn clusters) can be different each other and the size-effects are important for MR phenomenon [14]. If the sizes of the Mn clusters are much smaller than the mean-free path of spin up and spin down electrons in grain boundary region, the size-effect may be not taken into account temporarily (see Ref. [14], for details). In conclusion, the MR phenomenon in polycrystalline structure are studied by a simplified model in this Letter. We explain some experimental observations [1] qualitatively and try to discuss the physical origin of MR. The MR component at low temperatures is due to spin-dependent scattering by tiny Mn clusters existing in grain boundary regions, and the CMR component is caused by spin-fluctuation of clusters in grain regions. A higher Tc and resistivity of grain boundaries will degrade the CMR effect. Hence the grain boundary can offer a significant MR at low temperatures, but are not beneficial to huge MR in whole temperature region.
Acknowledgement This work was supported by the National Natural Science Foundation of China under grant No. 19774042.
References [1] X.W. Li, A. Gupta, G. Xiao, G.Q. Gong, Appl. Phys. Lett. 71 (1997) 1124; A. Gupta, G.Q. Gong, G. Xiao, P.R. Duncombe, P. Lecoeur, P. Trouilloud, Y.Y. Wang, V.P. David, J.Z. Sun, Phys. Rev. B 54 (1996) R15629. [2] W.-G. Yin, R. Tao, Phys. Rev. B 62 (2000) 550; W. Zhang, M. Zhuang, K. Xia, N. Ming, Phys. Lett. A 237 (1997) 90. [3] H.Y. Hwang, S.W. Cheong, N.P. Ong, B. Batlogg, Phys. Rev. Lett. 77 (1996) 2041. [4] L.J. Zou, H.Q. Lin, Q.Q. Zhang, J. Appl. Phys. 83 (1998) 7363. [5] M. Rubinstein, J. Appl. Phys. 87 (2000) 5019; M. Jaime, P. Lin, S.H. Chun, M.B. Salamon, P. Dorsey, M. Rubinstein, Phys. Rev. B 60 (1999) 1028. [6] H. Wu, Q.-q. Zheng, X.-g. Gong, Phys. Rev. B 61 (2000) 5217. [7] Y. Ju, C. Xu, Z. Li, J. Magn. Magn. Mater. 223 (2001) 267. [8] A. Milner, I.Ya. Korenblit, A. Gerber, Phys. Rev. B 60 (1999) 14821. [9] J.M.D. Coey, M. Viret, L. Ranno, K. Ounadjela, Phys. Rev. Lett. 75 (1995) 3910. [10] J.-Q. Wang, G. Xiao, Phys. Rev. B 50 (1994) 3423. [11] H.E. Cambloug, Phys. Rev. B 51 (1995) 16052. [12] P. Lyu, D.Y. Xing, J.M. Dong, J. Magn. Magn. Mater. 202 (1999) 405. [13] W.T. Doyle, J. Appl. Phys. 85 (1999) 2323. [14] A. Vedyayer, B. Mevel, N. Ryzhanora, M. Tshier, B. Dieny, J. Magn. Magn. Mater. 164 (1996) 91.
Physics Letters A 289 (2001) 106–110 www.elsevier.com/locate/pla
A study on magnetoresistivity of polycrystalline perovskite manganite films Yan Ju b , Zhen-Ya Li a,b,∗ a CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China b Department of Physics, Suzhou University, Suzhou 215006, China 1
Received 2 June 2001; accepted 28 August 2001 Communicated by J. Flouquet
Abstract Within the framework of effective medium theory and two-channel model, magnetoresistivity (MR) of polycrystalline perovskite manganite films are investigated. Grain boundaries are considered to result in the MR at low temperatures. Some features of recent experiments are explained qualitatively and the optimal choice of parameters to maximize MR in whole temperature region are discussed. 2001 Elsevier Science B.V. All rights reserved. PACS: 75.70.Pa; 75.70.Ak
1. Introduction The giant magnetoresistance effect (GMR) in magnetic granular alloys, considered to origin from asymmetrical scattering between spin-up and spin-down channels, has been studied thoroughly and has been found hard to meet application requirements well, such as huge resistivity change in the condition of low magnetic field B and high temperature T . Hence much attention was turned to other materials of intrinsically high degree of spin polarization, for example, the long researched doped perovskite manganite films. Recently, experiments [1] were made comparatively on epitaxial and polycrystalline perovskite manganite films (La0.67Sr0.33 MnO3 and La0.67Ca0.33 MnO3 ), * Corresponding author.
E-mail address: [email protected] (Z.-Y. Li). 1 Mailing address in China.
which demonstrated two features: (1) both of them exhibited a colossal magnetoresistivity effect (CMR) near critical temperature Tc , (2) contrasting to epitaxial films without MR at low temperatures (T Tc ), the MR in polycrystalline films increased almost linearly down to the lowest temperature. Offering significant MR over a wide temperature region, the polycrystalline perovskite manganite films were more attractive for applications, however theoretical researches on them were not enough. Though the CMR effect had been interpreted in some works [2], the mechanism of the MR in low temperature region was still confused. It was usually owned to the contribution of grain boundaries. But there was discrepancy in literature on the role of grain boundaries. It was proposed in Ref. [1] that spin-dependent scattering at grain boundaries, rather than magnetotunneling proposed by others [3], is responsible for the MR in low T region. In the case of qualities of carriers [4,5] and electronic structure [6], the doped perovskite manganite is
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 5 7 3 - 4
Y. Ju, Z.-Y. Li / Physics Letters A 289 (2001) 106–110
indeed a complicated system. However, its MR phenomenon has been studied by a model [2] extended from the treatment of magnetic granular alloys, which is based on such facts as: the transport in both systems is spin-dependent. Meanwhile, the low temperature behavior of MR in polycrystalline perovskite manganite films is much similar to the one in magnetic granular alloys [7,8]. It is nature to assume they due to the same mechanism of scattering on tiny ferromagnetic clusters as mentioned in Refs. [7,8]. Actually, tiny ferromagnetic Mn cluster may exist in perovskite manganite films due to local deviations from ferromagnetic order [9]. In this Letter, we present a model to study the polycrystalline perovskite manganite system, assuming the CMR effect be due to the near Tc mechanism inside grain regions, and the MR at low temperatures be due to the scattering by tiny Mn clusters existing in grain boundary regions, and experimental observations are interpreted qualitatively.
2. Formalism The transmission electron microscopy (TEM) of polycrystalline perovskite manganite films indicated that a well-defined grain morphology reminiscent of underlying polycrystalline SrTiO3 substrate, and furthermore, the wide-field Kerr microscope showed that the magnetic domains were defined by the grains [1]. Hence, we suppose that tiny ferromagnetic Mn clusters existing in grain regions of size of several µm are of strong coupling, such forming magnetic domains. However, in grain boundary regions, tiny ferromagnetic clusters are floppier than the bulk clusters [10] and of no interactions. Following spin-dependent twochannels model, MR is due to the asymmetric scattering in spin-up (+) and -down (−) channels, in which spins of carriers parallel and anti-parallel to the mag respectively. netic field B, Inside grain regions, when the spin of carrier is parallel to the magnetic moment (µ) of tiny Mn cluster, the carrier experiences weak scattering and hence a low resistivity (ρL ), otherwise, it experiences strong scattering and hence a high resistivity (ρH ). Taking a general spherical angle (θ, ϕ) between µ and B into account, the change of quantization axis from the magnetic moment direction to the magnetic field direction arises from a coordinate transformation characterized
by the corresponding rotation matrix cos(θ/2) sin(θ/2)e−iϕ . C= − sin(θ/2)eiϕ cos(θ/2)
107
(1)
In the simplified spin-dependent two-channel model, the two-point resistivity ρ(r , r ), which is forth-rank tensor, did not be taken into account, so that the resistivity ρ is a tensor of rank 2 [11] (see Ref. [11] for details). The bispinor ρL 0 ρ= 0 ρH transforms according to ρ = C−1 ρC, and after simple algebraic operations we got the usual form for resistiv± of grains as ity ρG ± = ρG
1 1 1 + cos(θ ) ± 1 − cos(θ ) 2 2
(2)
which is the diagonal terms of ρ . The nondiagonal terms of ρ average out with the azimuthal integration. ± Then taking the thermal average of ρG , 1 1 ± ρG (B, T ) = (ρH + ρL ) ∓ (ρH − ρL )L(α), 2 2
(3)
hence the conductivity σG± is σG± (B, T ) =
1
, ± ρG (B, T )
(4)
where ρH and ρL denotes the asymmetry of two channels. Note that the magnetoresistivity depends on the imbalance between the two conducting channels. If two channels are symmetrical, that is, ρH = ρL , no magnetoresistivity to be expected. The larger ρH /ρL , the larger magnetoresistivity is. In the following calculation, we take ρL = 1, ρH = 104 (arbit. unit) as Ref. [2]. L(α), equals to cos(θ ) , is Langevin function, with α = µBeff /KB T , and the effective magnetic field Beff includes the magnetic inter-cluster coupling described by a molecular field in addition to the external magnetic field [2]. The moment of the ferromagnetic Mn cluster depends on both the temperature and effective magnetic field, then we get 3Tc mG µB + , mG (B, T ) = L (5) Kb T T where mG is the reduced magnetization of grain regions.
108
Y. Ju, Z.-Y. Li / Physics Letters A 289 (2001) 106–110
Similarly, the resistivity ρB± of grain boundary regions, due to spin-dependent scattering on Mn cluster without coupling existing in the region of grain boundary, is treated quantum-mechanically and in the form of [7,8] J SZ J2 ± . ρB (B, T ) = ρ0 1 + 2 ∓ 2ρ0 (6) V V S Hence the conductivity σB± is σ ± (B, T )B =
1 , ρB± (B, T )
(7)
with reduced magnetization mB of grain boundary regions, (2S + 1)EH
SZ 2S + 1 = coth mB (B, T ) = S 2S kB T EH 1 coth − (8) . 2S kB T Here, V is the spin-independent potential and J is s–d exchange interaction, supposed to be smaller than V . ρ0 is the resistivity due to the scattering by potential V only. EH = gµB B/2 and g is the g factor of magnetic scatter taking value 2. In perovskite manganite films, carrier density in spin-up and -down channels is uneven, therefore the conductivities σG± and σB± must be revised as ± 2NG , (ρH + ρL ) ∓ (ρH − ρL )L(α) NB± , σB± (B, T ) = ρ0 (1 + J 2 /V 2 ) ∓ 2ρ0 (J /V )( SZ /S) (9)
σG± (B, T ) =
where N ± , the carrier density in each channel, is related with magnetic field and temperature and can be got easily from the expression of spin polarization of carrier [12] √ √ 3 1+m− 3 1−m N+ − N− =√ . P= + √ 3 N + N− 1+m+ 3 1−m It is obvious that N + = N − , in the case of P = 0, and ± NG (10) (B, T ) = 3 1 ± mG (B, T ) and NB± (B, T ) =
3
1 ± mB (B, T ).
(11)
Considering the polycrystalline films of island topology, i.e., region of grains embedded in a different host medium, namely the region of grain boundaries, ± of each channel is given the effective conductivity σeff in the form of Maxwell–Garnett equation [13] ± − σB± σeff
± σeff + 2σB±
= fG
σG± − σB±
σG± + 2σB±
,
(12)
where fG is the volume fraction of grain regions. If fG = 1, it returns to the case of epitaxial films. The total conductivity of σeff (B, T ) of the system is + − σeff (B, T ) = σeff (B, T ) + σeff (B, T ).
(13)
Then the MR of polycrystalline perovskite manganite film can be represented as follows: MR(T ) =
σeff (B, T ) − σeff (B = 0, T ) . σeff (B = 0, T )
(14)
3. Discussion and conclusion According to Eqs. (5) and (8)–(14), we derive the MR of polycrystalline perovskite manganite films over a wide T region and compare the calculated results with the experimental reports in Ref. [1]. In our calculation, we choose the magnetic moment of tiny cluster in grain regions µ = 20µB according to former theoretic studies [2], because the ferromagnetic clusters are found very small, and likewise the value of the total spin of cluster in grain boundary regions S = 13.5. Meanwhile, the value of ρ0 is taken larger than resistivity of grain region, such as 104 (arbit. unit), due to inhomogeneities of grain boundaries. The volume fraction of grain regions is selected as 0.9 and the magnetic field B as 4 T. In Fig. 1, the calculated MR in the case of polycrystalline films presents two components: (1) a steady incline of MR with increment of temperature in low T region, and (2) a sharp peak of MR near Tc , namely CMR effect. For comparison, we also deduce the case of epitaxial films by replacing fG = 1, that is, no grain boundary regions existing any more. There is only CMR effect left the near critical temperature where spin-fluctuation of ferromagnetic clusters is the largest, which is in agreement with former theoretic studies [2]. These results suggest that the CMR effect in doped perovskite manganite system be mainly
Y. Ju, Z.-Y. Li / Physics Letters A 289 (2001) 106–110
Fig. 1. The MR dependence on temperature for perovskite manganite films. (—) Epitaxial films, (– – –) polycrystalline films with the consideration of spin-polarization P , (– · – · –) polycrystalline films without the consideration of P .
caused by the spin-fluctuation of tiny clusters in grain regions. However, the MR in low T region is mainly caused by spin-dependent scattering on tiny clusters existing in grain boundary regions which are of no coupling and too small to be aligned by external magnetic field, hence the MR is enhanced continually till lowest temperature. In addition, the case of Tc = 230 K and Tc = 350 K are discussed, respectively, and find that, consistent with experimental reports [1] qualitatively, the CMR effect lessens with the increment of Tc . It maybe demonstrates why the shells of magnetic particles, according to the grain boundary regions in our model, could make a major contribution to GMR in magnetic granular alloys. It is an important presumption concluded from experiments and often cited in theoretic calculations. Because the Tc of magnetic metal is much larger than the above one, and it leads to a negligible CMR effect, but the spindependent scattering occurring at shells still results in large MR effect in low operating temperature. Note magnetoresistivity is an extra resistivity origining from the asymmetry of spin-down and -up channels which exhibits in two aspects. One is the imbalance of carrier density in both channels, indicated by the degree of spin polarization P of carriers. In Fig. 1, we have compared the cases with and without consideration of P . A larger MR will be got, if considered P . The other is connected with the degree of alignment of magnetic moments of ferromagnetic clusters (or particles). Both factors depends on temperature T and
109
Fig. 2. The MR dependence on temperature for different magnetic fields B.
Fig. 3. The MR and resistivity dependence on temperature for different spin-independent resistivity ρ0 .
It is shown in Fig. 2 that a stronger field B lead field B. to a sharper asymmetric broken in the system, hence a more manifest MR. Fig. 3 shows the MR and resistivity dependence on temperature for different spin-independent resistivity ρ0 of grain boundary region as 104 , 4 × 104 and 8 × 104 (arbit. unit), respectively. We find that grain boundary regions increase total resistivity of films and lead to the MR in low T region, but the CMR effect decreases simultaneously with the increment of ρ0 . Therefore, a high resistivity of films are not according to large MR in whole region of temperature. By the way, it is necessary to note that the calculated resistivity results cannot explain some experimental reports well, such as a sharp resistivity peak near Tc and the peak driven to high temperature with incre-
110
Y. Ju, Z.-Y. Li / Physics Letters A 289 (2001) 106–110
ment of magnetic field. It is possibly due to ignoring the qualities of carrier in our model, and more detail information on temperature and field dependence of ρH and ρL are needed which is the goal of our later work. On the other hand, in our model, it is assumed that the tiny ferromagnetic Mn clusters exist in the perovskite manganite films. However, the Mn cluster sizeeffect is neglected. In fact, the size of magnetic particles (Mn clusters) can be different each other and the size-effects are important for MR phenomenon [14]. If the sizes of the Mn clusters are much smaller than the mean-free path of spin up and spin down electrons in grain boundary region, the size-effect may be not taken into account temporarily (see Ref. [14], for details). In conclusion, the MR phenomenon in polycrystalline structure are studied by a simplified model in this Letter. We explain some experimental observations [1] qualitatively and try to discuss the physical origin of MR. The MR component at low temperatures is due to spin-dependent scattering by tiny Mn clusters existing in grain boundary regions, and the CMR component is caused by spin-fluctuation of clusters in grain regions. A higher Tc and resistivity of grain boundaries will degrade the CMR effect. Hence the grain boundary can offer a significant MR at low temperatures, but are not beneficial to huge MR in whole temperature region.
Acknowledgement This work was supported by the National Natural Science Foundation of China under grant No. 19774042.
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