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Effective medium theory of magnetoresistance in half-metallic granular systems
27 August 2001
Physics Letters A 287 (2001) 283–288 www.elsevier.com/locate/pla
Effective medium theory of magnetoresistance in half-metallic granular systems H. Sun b,∗ , Z.Y. 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 30 March 2001; accepted 22 June 2001 Communicated by J. Flouquet
Abstract We present a conductance network model to study magnetotransport properties of half-metallic granular systems. The transport process here is supposed to be controlled by both the charging energies and the spin-dependent tunneling between the grains. Using the effective medium theory, we obtain the conductivity of the system as a function of the temperature T and the applied magnetic field H . The effect of the charging energy and the spin polarization on the magnetoresistance is studied. It is also shown that the rapid decrease of the magnetoresistance with temperature may result from an unusual exponential decay of spin polarization as a consequence of the strong temperature dependence of grain surface magnetization. Our results are compared with the experiments on pressed powders of half-metallic CrO2 and a good agreement is found. 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Spin-polarized transport has been one of the foci in the contemporary condensed matter physics for both theoretical values and applicational prospects. In general, it will occur naturally in the materials for which the spin-up and spin-down electrons at the Fermi level are unequal in number. The usual case is ferromagnetic metals and their alloys, the spin-polarization of which is typically in the range of 25–40%. In the ultimate limit of complete spin polarization, one of the spin subbands is metallic, whereas the Fermi level falls into a gap of the other subband. Magnetic materials with
* Corresponding author.
E-mail address: [email protected] (Z.Y. Li). 1 Mailing address in China.
such band characteristics are named as half-metallic, which was first introduced by de Groot et al. [1] on the base of band structure calculations in NiMnSb and PtMnSb semi-Heusler phases. Since then, many materials have been predicted to have half-metallic properties, especially a series of metallic oxide ferromagnets, e.g., doped perovskite manganites, chromium dioxide, etc. Magnetic oxide materials having a high degree of spin polarization have attracted a great deal of attention since they are expected to exhibit enhanced spindependent transport phenomenon. Actually, an extrinsic large magnetoresistance (MR) effect at low temperature has successfully been observed in experiments on their polycrystalline samples, both in the form of thin films and bulk ceramics [2–5]. This type of magnetoresistance is usually ascribed to the spindependent tunneling between two grains separated by
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an insulating boundary. The electron tunnelling probability depends on the relative orientation of the magnetizations on neighboring grains, which can be substantially varied upon application of a magnetic field, thus leading to the so-called tunnel magnetoresistance (TMR). However, many of its characteristics are still confusing, particularly its rapid decay with increasing temperature. The similar effect is also reported in work of Coey et al. on pressed powder compacts of CrO2 [6]. Apart from the magnetoresistance decreasing exponentially with temperature, the experiments have also offered strong evidence of the relationship between the MR and the above tunnel mechanism. The compacts have resistivity about three orders of magnitude higher than the intrinsic value, which shows the resistance of the interparticle contact is dominant in the whole system. The average interparticle contact resistance is estimated as 60 k, much greater than the quantum limit RQ = h/2e2 . Moreover, the temperature behavior of the resistivity of the compacts can be fitted very well to the expression for granular metals, ρ ∝ exp (T0 /T )α , (1)
able. The percolation path may well describe another sample in Ref. [6], i.e., CrO2 powders diluted by insulating Cr2 O3 with the fraction near to the percolation concentration, but when it comes to the pure powders of CrO2 , there seems no enough evidence to support this idea. In fact, the theoretical value of MR is more than double the observed maximum data for the pure compacts (MR = 29% at T = 5 K) although it is near to that in the diluted sample (MR = 50% at T = 5 K). Besides, the qualitative explanation of the temperature dependence of the MR is also not satisfying. To study magnetotransport properties in halfmetallic granular systems further, we present in this Letter a conductance network model to describe these materials. The effect of charging energies and spin polarization on the conductivity and the TMR are investigated using the effective medium theory. The good agreement between our calculations and the experiments [6] suggests there may exist an unusual exponentially decay with temperature in spin polarization, which we believe comes from the effect of the surface magnetization of grains.
with α = 1/2, where T0 is proportional to the charging energy and the transport process is usually described as a thermally activated intergrain tunneling phenomenon of the charge carriers [7,8]. In order to explain the results of the experiments in [6], Coey proposed a model based on two main assumptions: (I) The current flows through weak conductance links in a percolation path and is controlled by the relative orientation of the magnetization of the half-metallic particles. As a consequence of this condition, the MR of the compacts is estimated to be 60%. (II) The CrO2 particles are sufficiently small for single-particle charging to influence the conductivity. In addition to the charging energy required to generate a pair of positively and negatively charged grains Ec , an extra charging energy Em of magnetic origin, which was first proposed by Helman and Abeles [7] to explain the MR of granular nickel, is introduced by the authors into the underlying transport mechanism. Then the rapid decay of the MR ratio is considered qualitatively to mainly result from the temperature dependence of Ec and Em . Investigating the above assumptions more carefully, it can be found that the validity of them is question-
2. The model As mentioned above, the typical α = 1/2 behavior of the resistivity obtained in Ref. [6] shows the system possesses the same transport mechanism as that in normal granular metals, where the transport process is controlled by the combination of tunneling through boundaries and the charging energy of grains. In the low electric field regime the charge carriers are thermally activated with number density ∼ exp(−Ec /kB T ), so the conductivity between two grains can be given as [8] σij = σijT exp(−Ec /kB T ),
(2)
where σijT refers to the tunneling mobility between the grains. It should be noted that here we do not include the contribution from Em since it is quite small as compared to Ec [9]. The MR effect in half-metallic granular systems is similar to that of the tunneling MR in the manganite tunnel junctions, hence it is expected to be interpreted in the same spin-dependent tunneling mechanism through the insulating layer in the tunnel junctions. Then the tunneling mobility between two grains
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σijT is, according to the conductance in the ferromagnetic metal–insulator–metal (FM/I/FM) junctions, given by [9] σijT ∝ 1 + P 2 cos Θ e−2κs , (3) where D↑ − D↓ P= D↑ + D↓ and κ=
(4)
2mφ/h¯ 2 .
(5)
Here, Θ is the angle between the magnetizations of two FMs, Dα (α = ↑, ↓) is the density of states at the Fermi energy EF for electrons with spin α, and s, m, and φ are the thickness of the barrier, the effective mass of electrons and barrier height, respectively. Substituting Eq. 3 into Eq. 2, the total conductivity, σij , between two grains i and j is written as σij = σ0 1 + P 2 cos Θ exp(−2κs − Ec /kB T ). (6) Neglecting the much smaller intrinsic resistance and the interaction between the grains, the whole system can be represented by a random conductance network in which each grain defines a site and two sites are linked together by σij . Applying the effective medium theory to the system, these random conductances are then replaced by a single average value, σe , which is equal to the conductivity of the network and determined implicitly by the equation [10] σij − σe (7) = 0. σij + 2σe Here the bracket . . . refers to the average over all pairs of grains. The distribution of Ec depends on that of the grain diameter d and of the barrier thickness s. Sheng [8] has pointed out that in order to ensure the homogeneity of the metallic grain concentration, the ratio d/s should have the same values for different regions in the system although both d and s may have a wide distribution. It follows that the product sEc is invariant for the same composition of the sample, which can be written as κsEc = c,
(8)
where κ and c are both constant. From Eq. (6) combined with this condition, it can be found that
there exists a certain value of s, c , sm = 2 2κ kB T
285
(9)
for which σij has a maximum at a given temperature. Now the average in Eq. (7) is determined by the distribution of s and Θ in the granular systems. We choose the probability distribution of s, P (s), as the form of s P (s) = 2 exp(−s/s0 ), (10) s0 and suppose the most probable value of s, s0 , is equal to sm . This function is an approximation to the one determined from electron micrographs [8]. Thin tunnel barriers, i.e., small s, correspond to large Ec that may block the transport process while thick tunnel barriers make tunneling difficult. From percolation theory [11], both the upper and the lower limit of √ s are dependent on temperature as proportional to 1/T . So we take 1 sm , 10 smax = 10sm . smin =
(11) (12)
Another simplifying assumption is made as follows. The system is assumed to consist of noninteractive superparamagnetic grains with the same magnetic moments µ. Of course, the grains in real systems are used to be single-domain ferromagnetic ones at low temperature. However, the magnetization curve in Ref. [6] shows quite a low coercivity and therefore a low anisotropy energy. So this assumption is acceptable only if we do not investigate the details of the conductivity behavior at low field. Then the probability distribution of the grain magnetization Mi at a given temperature T and applied field H is determined by F (Mi ) = exp(h cos θi ),
(13)
where h = µH /kB T and θi is the relative angle between the magnetization and the field. Taking the length of Mi as a unit, cos(Θ) can be expressed as cos(Θ) =
r2 − 1, 2
(14)
where r(r, θ, ϕ) is the vector sum of Mi and Mj . It is easily found that r is distributed within the radius of 2
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with the probability distribution Q(r) = exp(hr cos θ ).
(15)
Taking into account the factors mentioned above, Eq. (7) can be rewritten as smax 2 σij − σe ds r dr P (s) exp(hr) = 0, σij + 2σe
smin
(16)
−2
and the MR ratio is defined as σe (H ) − σe (0) . MR = σe (H )
(17)
Fig. 1. The temperature dependence of the reduced conductivity σ/σm with P = 1 for different values of c.
3. Results and discussions We have calculated the temperature dependence of the conductivity and of the magnetoresistance using Eqs. (16) and (17). The parameters are set as µ = 167µB and κ = 3 Å−1 to fit with experiments. Here, the value of the effective magnetic moment µ is reasonable for nanoparticles and the κ value has the same order of magnitude as that used for the ferromagnetic tunneling junctions [12]. The influence of c, corresponding to the charging energy Ec , and P , the spin polarization, is displayed in Figs. 1 and 2. In Fig. 1, the ratio of the conductivity of effective medium σe with P = 1 to its value at T = ∞, σm , is shown as a function of temperature for different values of c. It is found that the curves can be fitted very well to the law of the type σ (T ) ∝ exp(−(∆/T )1/2 ) and the value of ∆ is nearly proportional to c. Fig. 2 displays the temperature dependence of the calculated magnetoresistance for different c and P . It is shown that the MR ratio is not sensitive to the change of c except at very low temperature. From the inset of Fig. 2, we can see that when T < 50 K, the MR keeps rising with increasing c and this effect becomes more obvious while temperature drops. It is concluded that the small grains with high Coulomb energy Ec will lead to enhance the magnetoresistance at low temperature. Although there have been no much work on the relationship of the MR and the grain size in half-metallic systems, similar experiments for normal ferromagnetic granular metals [13] have reported the phenomena which can support our results.
Fig. 2. The temperature dependence of the MR for different P and c: (a) Three curves correspondent to P = 1 and c = 5, 1, 0.1, respectively. The inset gives the details of them at low temperature. (b) P = 0.8, c = 1. (c) P = 0.6, c = 1.
From Fig. 2, we can also see that spin-polarization P is a critical factor determining the MR and the larger P , the larger MR. When P = 1 and c = 1, the maximum value of MR reaches 43%, which is smaller than that obtained in Coey’s model (60%) or that from direct thermal average in Ref. [9] (50%), and closer to the value in the experiments (29%) for powders of CrO2 . Note that our calculated MR has a nearly linear dropping even when both the spin polarization and the charging energy are temperature invariant. This decreasing is due to the change of the magnetization distribution with temperature. In most cases, ideal half-metallic behavior is expected only at low temperature and the value of P usually decreases with increasing temperature. In order to investigate the relationship of P and the MR more deeply, we suppose that P decays exponentially
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287
with temperature from the half-metallic state (P = 1) at T = 0 K, P = exp(−T /T0 ).
(18)
Fig. 3 gives the MR as a function of T for c = 1 and different values of T0 . It is found that the MR exhibits temperature dependence in the same way as P does, MR = exp(−T /Tm ),
(19)
but at a higher speed (when T0 = 80 K, Tm = 46 K). The similar tendency has been found in the recent theoretical work on the MR for ferromagnetic junctions with half-metallic materials [14]. A good agreement between our results and the experiments [6] takes place when the value of T0 is chosen as 70 K (Figs. 4, 5). The corresponding field dependence of the resistance and the temperature-dependence of MR are give in Fig. 4 and Fig. 5, respectively, along with the experimental data from Ref. [6]. There are only two main deviations: one occurs for the resistance at high field, the other for MR at low temperature. In Fig. 4, the theoretical values of the resistance tend to saturate at high field while the experimental data exhibit a linearly dropping. This high-field tail may result from interfacial magnetization, which is not in the scheme of the present work. The MR at low temperature (MR = 41% at T = 5 K) is, as has been pointed out, closer to the observed values (MR = 29% at T = 5 K) compared to other previous work. However, it is still too large and we ascribe the discrepancy to the imperfection of tunnel barriers, e.g., the impurities in the boundary. It may be argued that the inelastic tunneling process involved in bulk- or surface-magnon emission or absorption can also contribute much to the temperature dependence of the MR. However, such a kind of tunneling will accompany sharp nonlinear features in the I –V curves even in the range of small voltage [15]. In the present case, no obvious nonlinear behavior has been observed in the I –V curve of the system [6], so we can conclude that the change of spin polarization is the most crucial factor. It is natural to expect the spin polarization P in half-metallic materials is more sensitive to temperature than normal ferromagnetic ones. The large Hund’s coupling existing in these materials makes their electronic structures intimately coupled to the magnetic structure, which may be very dynamic at elevated T ,
Fig. 3. The magnetoresistance curve with temperature for c = 1 and exponential decay of P as P = exp(−T /T0 ). (a) T0 = 80 K; (b) T0 = 50 K; (c) T0 = 20 K.
Fig. 4. Dependence of the resistance as a function of the magnetic field from the present calculation (line) and the experiments [6] (symbol) (c = 1, P = exp(−T /T0 ), T0 = 70 K).
Fig. 5. Magnetoresistance at c = 1 and P = exp(−T /T0 ) with T0 = 70 K (line), accompanied by experimental data from Ref. [6] (symbol).
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network model. The transport process in the system is described as thermal activated tunneling through grain boundaries which will lead to the typical α = 1/2 conductance law. It has been shown that both the charging energy Ec and the spin polarization P will influence the MR behavior. The rapid decrease of MR with temperature found in half-metallic granular systems is attributed to arise from the similar but slower decay of P which is determined by the surface magnetization.
Acknowledgement
Fig. 6. The comparison between the temperature dependence of spin polarization chosen in the Letter (line) and the surface magnetization Msurf vs. the normalized temperature τ core = T /Tccore from Ref. [17] (symbol).
This project was supported by the National Natural Science Foundation of China under Grant No. 19774042.
References and weaken the spin polarization. Data on resistivity and Hall effect in the Heusler alloys [16] show that P is strongly T -dependent and behaves roughly as the magnetization. Still, the form of P –T function used here is quite surprising for its decay is much more rapid than expected. We think this is because the grain surface state plays an central role in the tunneling between two grains and has an important effect on the spin-polarization. Usually, the surface magnetization Msurf decreases more rapidly than the core magnetization as the temperature increases, which leads to the same rapid drop of P . In order to examining the interrelation between P and Msurf , the P –T curve chosen here and the Msurf as a function of the reduced temperature τ core ≡ T /Tccore from Monte Carlo simulations in Ref. [17] are displayed simultaneously in Fig. 6. A similarity between them can be seen in this figure.
4. The summary To conclude, we have studied the magnetotransport properties for half-metallic granular systems using the effective medium theory on the base of a conductance
[1] R.A. de Groot, F.M. Muller, P.G. Van Engen, K.H. Buschow, Phys. Rev. Lett. 50 (1983) 2024. [2] H.Y. Hwang, S.-W. Cheong, M.P. Ong, B. Batlogg, Phys. Rev. Lett. 77 (1996) 2041. [3] H.Y. Hwang, S.-W. Cheong, Science 276 (1997) 1607. [4] A. Gupta, J.Z. Sun, J. Magn. Magn. Mater. 200 (1999) 24. [5] A. Gupta, X.W. Li, G. Xiao, J. Appl. Phys. 87 (2000) 6073. [6] J.M.D. Coey, A.E. Berkowitz, Ll. Balcells, F.F. Putris, Phys. Rev. Lett. 80 (1998) 3815; J.M.D. Coey, J. Appl. Phys. 85 (1999) 5576. [7] J.S. Helman, B. Abeles, Phys. Rev. Lett. 37 (1976) 1429. [8] P. Sheng, B. Abeles, Y. Arie, Phys. Rev. Lett. 31 (1973) 44. [9] J. Inoue, S. Maekawa, Phys. Rev. B 53 (1996) R11927. [10] S. Kirkpatrick, Phys. Rev. Lett. 27 (1971) 1722. [11] B. Sixou, J.P. Travers, J. Phys.: Condens. Matter 10 (1998) 593. [12] J.S. Moodera, L.R. Kinder, T.M. Wong, R. Meservey, Phys. Rev. Lett. 74 (1995) 3273. [13] T. Zhu, Y.J. Wang, Phys. Rev. B 60 (1999) 11918. [14] T. Ohsawa, J. Inoue, Phys. Rev. Lett. 84 (2000) 2501. [15] M. Hundley, J. Nickel, R. Ramesh, Y. Tokura, Invited talk at the MRS fall meeting in Boston, December 1–5, 1997. Published in MRS, Vol. 494, 1998. [16] M.J. Otto, H. Reihl, R.A.M. van Woerden, J. Wijingaard, P.J. van der Valk, C.F. van Bruggen, C. Haas, J. Magn. Magn. Mater. 70 (1987) 33. [17] H. Kachkachi, M. Nogues, E. Tronc, D.A. Garanin, J. Magn. Magn. Mater. 221 (2000) 158.
Physics Letters A 287 (2001) 283–288 www.elsevier.com/locate/pla
Effective medium theory of magnetoresistance in half-metallic granular systems H. Sun b,∗ , Z.Y. 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 30 March 2001; accepted 22 June 2001 Communicated by J. Flouquet
Abstract We present a conductance network model to study magnetotransport properties of half-metallic granular systems. The transport process here is supposed to be controlled by both the charging energies and the spin-dependent tunneling between the grains. Using the effective medium theory, we obtain the conductivity of the system as a function of the temperature T and the applied magnetic field H . The effect of the charging energy and the spin polarization on the magnetoresistance is studied. It is also shown that the rapid decrease of the magnetoresistance with temperature may result from an unusual exponential decay of spin polarization as a consequence of the strong temperature dependence of grain surface magnetization. Our results are compared with the experiments on pressed powders of half-metallic CrO2 and a good agreement is found. 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Spin-polarized transport has been one of the foci in the contemporary condensed matter physics for both theoretical values and applicational prospects. In general, it will occur naturally in the materials for which the spin-up and spin-down electrons at the Fermi level are unequal in number. The usual case is ferromagnetic metals and their alloys, the spin-polarization of which is typically in the range of 25–40%. In the ultimate limit of complete spin polarization, one of the spin subbands is metallic, whereas the Fermi level falls into a gap of the other subband. Magnetic materials with
* Corresponding author.
E-mail address: [email protected] (Z.Y. Li). 1 Mailing address in China.
such band characteristics are named as half-metallic, which was first introduced by de Groot et al. [1] on the base of band structure calculations in NiMnSb and PtMnSb semi-Heusler phases. Since then, many materials have been predicted to have half-metallic properties, especially a series of metallic oxide ferromagnets, e.g., doped perovskite manganites, chromium dioxide, etc. Magnetic oxide materials having a high degree of spin polarization have attracted a great deal of attention since they are expected to exhibit enhanced spindependent transport phenomenon. Actually, an extrinsic large magnetoresistance (MR) effect at low temperature has successfully been observed in experiments on their polycrystalline samples, both in the form of thin films and bulk ceramics [2–5]. This type of magnetoresistance is usually ascribed to the spindependent tunneling between two grains separated by
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H. Sun, Z.Y. Li / Physics Letters A 287 (2001) 283–288
an insulating boundary. The electron tunnelling probability depends on the relative orientation of the magnetizations on neighboring grains, which can be substantially varied upon application of a magnetic field, thus leading to the so-called tunnel magnetoresistance (TMR). However, many of its characteristics are still confusing, particularly its rapid decay with increasing temperature. The similar effect is also reported in work of Coey et al. on pressed powder compacts of CrO2 [6]. Apart from the magnetoresistance decreasing exponentially with temperature, the experiments have also offered strong evidence of the relationship between the MR and the above tunnel mechanism. The compacts have resistivity about three orders of magnitude higher than the intrinsic value, which shows the resistance of the interparticle contact is dominant in the whole system. The average interparticle contact resistance is estimated as 60 k, much greater than the quantum limit RQ = h/2e2 . Moreover, the temperature behavior of the resistivity of the compacts can be fitted very well to the expression for granular metals, ρ ∝ exp (T0 /T )α , (1)
able. The percolation path may well describe another sample in Ref. [6], i.e., CrO2 powders diluted by insulating Cr2 O3 with the fraction near to the percolation concentration, but when it comes to the pure powders of CrO2 , there seems no enough evidence to support this idea. In fact, the theoretical value of MR is more than double the observed maximum data for the pure compacts (MR = 29% at T = 5 K) although it is near to that in the diluted sample (MR = 50% at T = 5 K). Besides, the qualitative explanation of the temperature dependence of the MR is also not satisfying. To study magnetotransport properties in halfmetallic granular systems further, we present in this Letter a conductance network model to describe these materials. The effect of charging energies and spin polarization on the conductivity and the TMR are investigated using the effective medium theory. The good agreement between our calculations and the experiments [6] suggests there may exist an unusual exponentially decay with temperature in spin polarization, which we believe comes from the effect of the surface magnetization of grains.
with α = 1/2, where T0 is proportional to the charging energy and the transport process is usually described as a thermally activated intergrain tunneling phenomenon of the charge carriers [7,8]. In order to explain the results of the experiments in [6], Coey proposed a model based on two main assumptions: (I) The current flows through weak conductance links in a percolation path and is controlled by the relative orientation of the magnetization of the half-metallic particles. As a consequence of this condition, the MR of the compacts is estimated to be 60%. (II) The CrO2 particles are sufficiently small for single-particle charging to influence the conductivity. In addition to the charging energy required to generate a pair of positively and negatively charged grains Ec , an extra charging energy Em of magnetic origin, which was first proposed by Helman and Abeles [7] to explain the MR of granular nickel, is introduced by the authors into the underlying transport mechanism. Then the rapid decay of the MR ratio is considered qualitatively to mainly result from the temperature dependence of Ec and Em . Investigating the above assumptions more carefully, it can be found that the validity of them is question-
2. The model As mentioned above, the typical α = 1/2 behavior of the resistivity obtained in Ref. [6] shows the system possesses the same transport mechanism as that in normal granular metals, where the transport process is controlled by the combination of tunneling through boundaries and the charging energy of grains. In the low electric field regime the charge carriers are thermally activated with number density ∼ exp(−Ec /kB T ), so the conductivity between two grains can be given as [8] σij = σijT exp(−Ec /kB T ),
(2)
where σijT refers to the tunneling mobility between the grains. It should be noted that here we do not include the contribution from Em since it is quite small as compared to Ec [9]. The MR effect in half-metallic granular systems is similar to that of the tunneling MR in the manganite tunnel junctions, hence it is expected to be interpreted in the same spin-dependent tunneling mechanism through the insulating layer in the tunnel junctions. Then the tunneling mobility between two grains
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σijT is, according to the conductance in the ferromagnetic metal–insulator–metal (FM/I/FM) junctions, given by [9] σijT ∝ 1 + P 2 cos Θ e−2κs , (3) where D↑ − D↓ P= D↑ + D↓ and κ=
(4)
2mφ/h¯ 2 .
(5)
Here, Θ is the angle between the magnetizations of two FMs, Dα (α = ↑, ↓) is the density of states at the Fermi energy EF for electrons with spin α, and s, m, and φ are the thickness of the barrier, the effective mass of electrons and barrier height, respectively. Substituting Eq. 3 into Eq. 2, the total conductivity, σij , between two grains i and j is written as σij = σ0 1 + P 2 cos Θ exp(−2κs − Ec /kB T ). (6) Neglecting the much smaller intrinsic resistance and the interaction between the grains, the whole system can be represented by a random conductance network in which each grain defines a site and two sites are linked together by σij . Applying the effective medium theory to the system, these random conductances are then replaced by a single average value, σe , which is equal to the conductivity of the network and determined implicitly by the equation [10] σij − σe (7) = 0. σij + 2σe Here the bracket . . . refers to the average over all pairs of grains. The distribution of Ec depends on that of the grain diameter d and of the barrier thickness s. Sheng [8] has pointed out that in order to ensure the homogeneity of the metallic grain concentration, the ratio d/s should have the same values for different regions in the system although both d and s may have a wide distribution. It follows that the product sEc is invariant for the same composition of the sample, which can be written as κsEc = c,
(8)
where κ and c are both constant. From Eq. (6) combined with this condition, it can be found that
there exists a certain value of s, c , sm = 2 2κ kB T
285
(9)
for which σij has a maximum at a given temperature. Now the average in Eq. (7) is determined by the distribution of s and Θ in the granular systems. We choose the probability distribution of s, P (s), as the form of s P (s) = 2 exp(−s/s0 ), (10) s0 and suppose the most probable value of s, s0 , is equal to sm . This function is an approximation to the one determined from electron micrographs [8]. Thin tunnel barriers, i.e., small s, correspond to large Ec that may block the transport process while thick tunnel barriers make tunneling difficult. From percolation theory [11], both the upper and the lower limit of √ s are dependent on temperature as proportional to 1/T . So we take 1 sm , 10 smax = 10sm . smin =
(11) (12)
Another simplifying assumption is made as follows. The system is assumed to consist of noninteractive superparamagnetic grains with the same magnetic moments µ. Of course, the grains in real systems are used to be single-domain ferromagnetic ones at low temperature. However, the magnetization curve in Ref. [6] shows quite a low coercivity and therefore a low anisotropy energy. So this assumption is acceptable only if we do not investigate the details of the conductivity behavior at low field. Then the probability distribution of the grain magnetization Mi at a given temperature T and applied field H is determined by F (Mi ) = exp(h cos θi ),
(13)
where h = µH /kB T and θi is the relative angle between the magnetization and the field. Taking the length of Mi as a unit, cos(Θ) can be expressed as cos(Θ) =
r2 − 1, 2
(14)
where r(r, θ, ϕ) is the vector sum of Mi and Mj . It is easily found that r is distributed within the radius of 2
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with the probability distribution Q(r) = exp(hr cos θ ).
(15)
Taking into account the factors mentioned above, Eq. (7) can be rewritten as smax 2 σij − σe ds r dr P (s) exp(hr) = 0, σij + 2σe
smin
(16)
−2
and the MR ratio is defined as σe (H ) − σe (0) . MR = σe (H )
(17)
Fig. 1. The temperature dependence of the reduced conductivity σ/σm with P = 1 for different values of c.
3. Results and discussions We have calculated the temperature dependence of the conductivity and of the magnetoresistance using Eqs. (16) and (17). The parameters are set as µ = 167µB and κ = 3 Å−1 to fit with experiments. Here, the value of the effective magnetic moment µ is reasonable for nanoparticles and the κ value has the same order of magnitude as that used for the ferromagnetic tunneling junctions [12]. The influence of c, corresponding to the charging energy Ec , and P , the spin polarization, is displayed in Figs. 1 and 2. In Fig. 1, the ratio of the conductivity of effective medium σe with P = 1 to its value at T = ∞, σm , is shown as a function of temperature for different values of c. It is found that the curves can be fitted very well to the law of the type σ (T ) ∝ exp(−(∆/T )1/2 ) and the value of ∆ is nearly proportional to c. Fig. 2 displays the temperature dependence of the calculated magnetoresistance for different c and P . It is shown that the MR ratio is not sensitive to the change of c except at very low temperature. From the inset of Fig. 2, we can see that when T < 50 K, the MR keeps rising with increasing c and this effect becomes more obvious while temperature drops. It is concluded that the small grains with high Coulomb energy Ec will lead to enhance the magnetoresistance at low temperature. Although there have been no much work on the relationship of the MR and the grain size in half-metallic systems, similar experiments for normal ferromagnetic granular metals [13] have reported the phenomena which can support our results.
Fig. 2. The temperature dependence of the MR for different P and c: (a) Three curves correspondent to P = 1 and c = 5, 1, 0.1, respectively. The inset gives the details of them at low temperature. (b) P = 0.8, c = 1. (c) P = 0.6, c = 1.
From Fig. 2, we can also see that spin-polarization P is a critical factor determining the MR and the larger P , the larger MR. When P = 1 and c = 1, the maximum value of MR reaches 43%, which is smaller than that obtained in Coey’s model (60%) or that from direct thermal average in Ref. [9] (50%), and closer to the value in the experiments (29%) for powders of CrO2 . Note that our calculated MR has a nearly linear dropping even when both the spin polarization and the charging energy are temperature invariant. This decreasing is due to the change of the magnetization distribution with temperature. In most cases, ideal half-metallic behavior is expected only at low temperature and the value of P usually decreases with increasing temperature. In order to investigate the relationship of P and the MR more deeply, we suppose that P decays exponentially
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with temperature from the half-metallic state (P = 1) at T = 0 K, P = exp(−T /T0 ).
(18)
Fig. 3 gives the MR as a function of T for c = 1 and different values of T0 . It is found that the MR exhibits temperature dependence in the same way as P does, MR = exp(−T /Tm ),
(19)
but at a higher speed (when T0 = 80 K, Tm = 46 K). The similar tendency has been found in the recent theoretical work on the MR for ferromagnetic junctions with half-metallic materials [14]. A good agreement between our results and the experiments [6] takes place when the value of T0 is chosen as 70 K (Figs. 4, 5). The corresponding field dependence of the resistance and the temperature-dependence of MR are give in Fig. 4 and Fig. 5, respectively, along with the experimental data from Ref. [6]. There are only two main deviations: one occurs for the resistance at high field, the other for MR at low temperature. In Fig. 4, the theoretical values of the resistance tend to saturate at high field while the experimental data exhibit a linearly dropping. This high-field tail may result from interfacial magnetization, which is not in the scheme of the present work. The MR at low temperature (MR = 41% at T = 5 K) is, as has been pointed out, closer to the observed values (MR = 29% at T = 5 K) compared to other previous work. However, it is still too large and we ascribe the discrepancy to the imperfection of tunnel barriers, e.g., the impurities in the boundary. It may be argued that the inelastic tunneling process involved in bulk- or surface-magnon emission or absorption can also contribute much to the temperature dependence of the MR. However, such a kind of tunneling will accompany sharp nonlinear features in the I –V curves even in the range of small voltage [15]. In the present case, no obvious nonlinear behavior has been observed in the I –V curve of the system [6], so we can conclude that the change of spin polarization is the most crucial factor. It is natural to expect the spin polarization P in half-metallic materials is more sensitive to temperature than normal ferromagnetic ones. The large Hund’s coupling existing in these materials makes their electronic structures intimately coupled to the magnetic structure, which may be very dynamic at elevated T ,
Fig. 3. The magnetoresistance curve with temperature for c = 1 and exponential decay of P as P = exp(−T /T0 ). (a) T0 = 80 K; (b) T0 = 50 K; (c) T0 = 20 K.
Fig. 4. Dependence of the resistance as a function of the magnetic field from the present calculation (line) and the experiments [6] (symbol) (c = 1, P = exp(−T /T0 ), T0 = 70 K).
Fig. 5. Magnetoresistance at c = 1 and P = exp(−T /T0 ) with T0 = 70 K (line), accompanied by experimental data from Ref. [6] (symbol).
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network model. The transport process in the system is described as thermal activated tunneling through grain boundaries which will lead to the typical α = 1/2 conductance law. It has been shown that both the charging energy Ec and the spin polarization P will influence the MR behavior. The rapid decrease of MR with temperature found in half-metallic granular systems is attributed to arise from the similar but slower decay of P which is determined by the surface magnetization.
Acknowledgement
Fig. 6. The comparison between the temperature dependence of spin polarization chosen in the Letter (line) and the surface magnetization Msurf vs. the normalized temperature τ core = T /Tccore from Ref. [17] (symbol).
This project was supported by the National Natural Science Foundation of China under Grant No. 19774042.
References and weaken the spin polarization. Data on resistivity and Hall effect in the Heusler alloys [16] show that P is strongly T -dependent and behaves roughly as the magnetization. Still, the form of P –T function used here is quite surprising for its decay is much more rapid than expected. We think this is because the grain surface state plays an central role in the tunneling between two grains and has an important effect on the spin-polarization. Usually, the surface magnetization Msurf decreases more rapidly than the core magnetization as the temperature increases, which leads to the same rapid drop of P . In order to examining the interrelation between P and Msurf , the P –T curve chosen here and the Msurf as a function of the reduced temperature τ core ≡ T /Tccore from Monte Carlo simulations in Ref. [17] are displayed simultaneously in Fig. 6. A similarity between them can be seen in this figure.
4. The summary To conclude, we have studied the magnetotransport properties for half-metallic granular systems using the effective medium theory on the base of a conductance
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