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Temperature dependence of the giant magnetoresistance in Fe–Cr multilayers—Intralayer and interlayer exchange energies
Journal of Magnetism and Magnetic Materials 323 (2011) 646–649
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Temperature dependence of the giant magnetoresistance in Fe–Cr multilayers—Intralayer and interlayer exchange energies R.S. Patel a, A.K. Majumdar b,n a b
Birla Institute of Technology and Science-Pilani, Goa Campus, Zuarinagar, Goa 403726, India S N Bose National Centre for Basic Sciences, Sector III, Block JD, Kolkata 700 098, West Bengal, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 September 2010 Received in revised form 21 October 2010 Available online 28 October 2010
We present field and temperature dependence data on giant magnetoresistive (GMR) ion-beam sputtered Fe–Cr multilayers of varying Cr thickness. We show that the decrease in GMR with temperature is related to the decrease in sublattice magnetization due to thermal excitation of magnons in the antiferromagnetic configuration. The intralayer and the interlayer exchange energies thus obtained vary systematically as the Cr thickness increases. The corresponding decrease in the measured saturation field further supports our interpretation leading to a better understanding of the physics of GMR. & 2010 Elsevier B.V. All rights reserved.
Keywords: GMR Fe-Cr multilayers Exchange energies
1. Introduction The research on magnetism till late 20th century focused on its basic understanding and also on applications as soft and hard magnetic materials, both in crystalline and amorphous forms. In 1980s it was observed that surface and interface properties deviate considerably from those of the bulk. Then came, for the first time, the industrial application of electronic properties which depend on the spin of the electrons. In 1986 exchange coupling between ferromagnetic thin films across a non-magnetic metallic interlayer was realized in Fe–Cr and Co–Cu structures and rare earth-based multilayers giving rise to the so-called Giant Magnetoresistance (GMR¼(((r(T,H)–r(T,0))/r(T,0)) 100%)). Here the electrical resistivity decreased by almost 50% on the application of a magnetic field of a few kOe and the decrease was attributed to the spindependent bulk and interface scattering. The exchange coupling ¨ between Fe layers was established by Grunberg et al. [1] through scattering of light by spin waves. Visscher and Zhang [2] had shown that in the ferromagnetic configuration, realized by applying a magnetic field, the electrical resistivity decreases (negative GMR) from its value at zero or low fields when the Fe layers are antiferromagnetically aligned for appropriate Cr thickness. They computed the bulk scattering using the 2-current conduction model of Fert and Campbell [3] and found that the majority band of Fe has a much larger resistivity than the minority band. Butler et al. [4] used density functional theory for calculating the electronic structure and conductivity in many GMR systems. They
n
Corresponding author. E-mail address: [email protected] (A.K. Majumdar).
0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.10.036
found for Fe–Cr that the majority spin electrons at the Fermi level would see large difference between the atomic potentials of Fe and Cr and experience strong reflections from interfaces as well as strong impurity scattering at interdiffused Fe–Cr zones. Their calculations also showed that the density of states (DOS) at the mismatched spin channel was large. So the resistivity was much larger for the majority band due to both potential mismatch and higher DOS. A recent book [5] gives an excellent review on the physics and applications of GMR. Giant magnetoresistance (GMR) depends on various parameters of the multilayer structure, e.g. spacer layer thickness, interlayer surface roughness, etc. Petroff et al. [6] found that the GMR oscillated with the Cr thickness but varied rather slowly with the thickness of Fe. The ratio of the Cr thickness to the electron mean free path played a major role in determining the GMR. The scattering by Cr impurities in Fe layer was found to be less effective than the scattering by interface roughness. The scattering by Cr impurities was certainly spin dependent but probably less than the interface scattering. Varying interface roughness could considerably change GMR. There are many factors that might affect the temperature dependence of GMR. Thermal demagnetization of sublattice magnetization and spin-flip scattering by collision with spin waves are the few important factors. Hasegawa [7] studied the GMR in magnetic multilayers at finite temperatures with the use of the functional-integral method in the static approximation. The effect of spin fluctuation was shown to play an important role in the temperature dependence of the GMR. The temperature dependence of the calculated GMR was much stronger than that of the magnetization and it showed an almost linear decrease near the Curie temperature when the temperature was raised.
R.S. Patel, A.K. Majumdar / Journal of Magnetism and Magnetic Materials 323 (2011) 646–649
647
In an earlier work [8], we had investigated, among many other things, the temperature dependence of resistivity and magnetoresistance of three samples of Xe-ion-beam sputtered 30 bi-layers of ˚ ˚ Most importantly, the nearly fixed thickness Fe(20 A)/Cr( 12 A). sublattice magnetization m(T), a measure of the aniferromagnetic (AF) order parameter in the multilayer, was calculated using a simplified Hubbard model in terms of the planar and interlayer exchange energies. We found that the additional spin-dependent resistivity in the AF configuration (H¼0), proportional to m(T), decreased with increasing temperature in a rather complicated fashion but much the same way as our experimental data of the ˚ ˚ sample. Fe(20 A)/Cr( 12 A) The motivation behind the present work is to understand the inplane and out-of-plane exchanges, in Fe–Cr multilayers of variable ˚ keeping Fe fixed at 20 A˚ through the temCr thickness (6–14 A) perature dependence of the GMR. We find that the intralayer and the interlayer exchange energies, obtained by fitting the temperature dependence of the GMR to our earlier theory [8], are consistent with the Cr thickness variation of the measured Hsat, the field at which the GMR (H) saturates.
2. Experimental Details ˚ ˚ 30 bi-layers of Fe–Cr of composition Si/Cr(50 A)/[Fe(20 A)/ ˚ 30/Cr(50-t (A)), ˚ Cr(t A)] where t ¼6, 8, 10, 12 and 14 A˚ were grown on Si substrates by ion-beam sputter deposition technique [9] using 1100 eV Xe ions of 30 mA beam current. Electrical resistance was measured using the standard 4-probe dc-method with a resolution of 3 parts in 106. However, the resistivity values were accurate to only within 5–10% due to uncertainties in the measurements of the sample dimensions. The typical dimensions of the samples for the measurements were 3 8 mm2 and the current used was 10 mA. Magnetic field, provided by a Varian 15 in. (Model V-3800) electromagnet was always applied in the plane of the sample parallel to the current. The in-plane magnetoresistance (MR) was measured at every 100 mK with a stability of 10 mK from 2.3 to 300 K up to H¼16 kOe.
3. Results and discussion Fig. 1 shows magnetoresistance (MR) versus H plots at 4.2 and ˚ The MR saturates around 11 kOe. 300 K for sample 2 (Cr ¼8 A). Similar plots give Hsat ¼11, 11, 8 and 5.5 kOe for samples 2–5 with ˚ respectively. The samples on the average Cr ¼8, 10, 12 and 14 A, have residual resistivity of 30 mO cm, Residual Resistivity Ratio (RRR¼R(300 K)/R (4.2 K)) of 1.8 at Hsat and GMR of 32% at 4.2 K. Small angle X-ray reflectivity measurements [10] quantitatively ˚ show that the roughness of both Fe and Cr layers is (5.570.5) A. The grazing incidence X-ray diffraction (GIXD) measurements [10] on these samples have been performed by keeping the angle of incidence fixed at around yi E11 and making the detector scan over a large angular region. All the samples, in the limited angular region from 40 to 501, exhibit broad diffraction maxima at around 2y E44.51. No crystalline peaks for Fe or Cr are observed over the entire scan. The existence of broad peaks signifies that all the multilayer samples are polycrystalline in nature and the estimated grain size is found to be of the order of 15 nm. Before presenting our data on the temperature dependence of GMR, let us summarize the theory we are going to use in its interpretation. The GMR in multilayers is mainly due to the different magnetic configurations in H¼0 and H¼Hsat state. The temperature dependence of the GMR depends upon the number of magnons in the two states. We can very well assume that the temperature dependence of the phonon contribution is the same in
˚ at 4.2 Fig. 1. Magnetoresistance (MR) versus magnetic field H for sample 2 (Cr¼8 A) and 300 K. Black and red points are the data points in magnetic field increment and decrement cycles showing some low-field hysteresis. HSat 11 kOe for this sample. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
these two states. Thermal excitation of magnons, which causes local spin disorder in multilayers, will be different in the two configurations. We observe that the net magnetization changes very little (1% till 100 K and 5% till 300 K) in the presence of Hsat, i.e., the sublattice magnetization in the FM configuration is almost constant with temperature. For thermal excitation of spin waves in highly anisotropic antiferromagnets with weak interlayer antiferromagnetic coupling between the planes, Singh et al. [11] found that at low temperatures /nSTpT2. The extension of this theory in the present case, where each Fe layer is ferromagnetic and is antiferromagnetically coupled to the neighboring Fe layer, in terms of the planer (intralayer) and interlayer exchange energies, Jp and Jz, respectively, gives [8]
DrðTÞ 1 T ¼ 1 2 Drð0Þ p Jp
Z p=2 0
dqz ln
1 1eJz =T ð1cos2 qz Þ1=2
! ð1Þ
where qz is an interlayer magnon vector. This equation shows that the MR decreases approximately as T2 at low temperatures (T5Jz) and there is a cross-over to an approximately linear (T ln T) behavior at higher temperatures (T bJz). The function 1 (T2 or T ln T) has always a convex upward curvature. In Fig. 2 we have plotted Dr ¼ r(T, H¼0) r(T, H¼Hsat) versus T data for sample 3. The temperature ranges in the analysis are till 150, 200, 250 and 300 K. The solid lines are the least-squares fitted curves for fits to Eq. (1). It can be seen that the fit is excellent for the range To150 K. The fit becomes worse for higher temperature ranges. It is also clearly seen that Dr(T) has concave upward curvature near 300 K. The reason could be as follows: in the derivation of Dr(T) it is assumed that the sublattice magnetization in the FM configuration of the multilayers is almost constant with temperature but the change in magnetization in the AF configuration is huge and this is the governing factor for Dr(T). But in reality there is some change in sublattice magnetization with temperature in the FM configuration especially at high temperatures. Magnetization changes by 1% till 100 K but by 5% till 300 K. This change in magnetization in the FM configuration leads to the observed deviation from the predicted functional dependence of Dr(T), as given by Eq. (1). For this reason we restricted our analysis up to only 150 K for finding out some meaningful values of the parameters, Jp and Jz. The integral in Eq.(1) was evaluated numerically and used in a
648
R.S. Patel, A.K. Majumdar / Journal of Magnetism and Magnetic Materials 323 (2011) 646–649
Fig. 2. Dr ¼ r(T, H¼ 0)–r(T, H¼ Hsat) versus temperature (T) data for sample 3. The temperature ranges in the analysis are till 150, 200, 250 and 300 K. The solid lines are the least-squares fitted curves for fits to Eq. (1).
Table 1 Values of w2, correlation coefficient R2, the parameters Jp and Jz from fits to Eq. (1) till 150 K for samples 2, 3, 4 and 5.
Fig. 3. Dr ¼ r(T, H ¼0)–r(T, H¼Hsat) versus temperature (T) data for samples 2, 3, 4 and 5. The solid lines are the least-squares fitted curves for fits to Eq. (1).
3-parameter least squares fit program. Fig. 3 shows the best-fit curves of Dr(T) data to Eq. (1) till 150 K for samples 2, 3, 4 and 5. The values of w2 are consistent with the experimental resolution and the correlation coefficient R2 40.99999. The values of w2, R2 and the parameters, Jp and Jz are presented in Table 1. The parameter Jp that represents the planar/ intralayer exchange energy is much below the Curie temperature (1040 K) of bulk Fe. However, it may be closer to the unknown Curie temperature of 20 A˚ thick Fe film. The parameter Jz, the interlayer exchange energy, is found to be rather close to the spin-glass transition temperatures derived from the low-field magnetization and acsusceptibility measurements [12]. From Table 1 we find that Jz decreases gradually from 117 to 45 K and Jp increases from 201 to 330 K as Cr thickness increases from 8 A˚ (sample 2) to 14 A˚ (sample 5).
Samples
w2 (10 6)
R2
Jp (K)
Jz (K)
2 3 4 5
4.45 5.43 3.55 12.90
0.999996 0.999995 0.999996 0.999987
2017 7 213 7 8 265 7 6 3307 16
117 76 102 76 76 75 45 75
This is because of the decrease in the antiferromagnetic coupling between the Fe layers as seen from the corresponding decrease of the measured Hsat from 11 to 5.5 kOe. In an earlier work on the temperature dependence of the GMR Suzuki and Taga [13] found that Dr(T)¼ r(H¼0, T)–r(H¼Hsat, T) varies as T3/2 in the temperature range 2rT r300 K for Co/Cu superlattice. Their interpretation was based on the spin-mixing concept, viz., with increasing temperature spin-up and spin-down channels could not be assumed to be independent and there would be spin mixing. Dr(T) depended on the difference in population between the majority and minority spin electrons at T. With increasing temperature, this population difference decreased and so did Dr(T). Mattson et al. [14] found in magnetron sputtered Fe–Cr superlattice that the magnetoresistivity of the antiferromagnetically coupled films fell from its maximum value at low temperatures with a T2 dependence, while a ferromagnetically (FM) coupled film showed a T3/2 behavior. These power laws were explained as a consequence of the thermal excitation of magnons. Camley and Barnas [15] had suggested that the temperature dependence of the GMR could be attributed to the variation of the mean free path (MFP) with temperature.
4. Conclusions To conclude, the temperature dependence of the GMR is satisfactorily explained in terms of the thermal excitation of spin
R.S. Patel, A.K. Majumdar / Journal of Magnetism and Magnetic Materials 323 (2011) 646–649
waves in the ferromagnetic Fe layers with weak interlayer antiferromagnetic coupling between the neighboring Fe layers. We also show that the planar/intralayer exchange energy, Jp increases with increasing Cr thickness while the interlayer exchange energy, Jz decreases with it. The corresponding decrease in the measured Hsat, the saturation field, further supports this interpretation and strengthens the physics behind GMR.
Acknowledgements We acknowledge D. Temple and C. Pace of MCNC of North Carolina, USA, for providing the samples and the Department of Science and Technology, Government of India, for financial assistance (Project no. SR/S2/CMP-18/204). References ¨ [1] P. Grunberg, R. Schreiber, Y. Pang, M.B. Brodsky, H. Sowers, Phys. Rev. Lett. 57 (1986) 2442.
649
[2] P.B. Visscher, Hui Zhang, Phys. Rev. B 48 (1993) 6672. [3] A. Fert, I.A. Campbell, J. Phys. F: Metal Phys. 6 (1976) 849. [4] W.H. Butler, X.-G. Zhang, D.M.C. Nicholson, J.M. MacLaren, J. Magn. Magn. Mater. 151 (1995) 354. [5] Uwe Hartmann, (Ed.), Magnetic multilayers and giant magnetoresistance: fundamentals and industrial applications, Springer Series in Surface Sciences, vol. 37, Springer, Berlin, 1999. [6] F. Petroff, A. Barthe´le´my, A. Hamzic, A. Fert, P. Etienne, S. Lequien, G. Creuzet, J. Magn. Magn. Mater. 93 (1991) 95. [7] H. Hasegawa, Phys. Rev. B 47 (1993) 15080. [8] A.K. Majumdar, A.F. Hebard, A. Singh, D. Temple, Phys. Rev. B 65 (2002) 054408. [9] R.S. Patel, A.K. Majumdar, A.F. Hebard, D. Temple, J. Appl. Phys. 97 (2005) 033910. [10] P. Khatua, Ph.D. Thesis, Indian Institute of Technology, Kanpur, India, 2006, Chapter 3. [11] A. Singh, Z. Teˇsanovic´, H. Tang, G. Xiao, C.L. Chien, J.C. Walker, Phys. Rev. Lett. 64 (1990) 2571. [12] R.S. Patel, A.K. Majumdar, A.K. Nigam, D. Temple, C. Pace, J. Appl. Phys. 100 (2006) 123914. [13] M. Suzuki, Y. Taga, Phys. Rev. B 52 (1995) 361. [14] J.E. Mattson, M.E. Brubaker, C.H. Sowers, M. Conover, Z. Qiu, S.D. Bader, Phys. Rev. B 44 (1991) 9378. [15] R.E. Camley, J. Barnas, Phys. Rev. Lett. 63 (1989) 664.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Temperature dependence of the giant magnetoresistance in Fe–Cr multilayers—Intralayer and interlayer exchange energies R.S. Patel a, A.K. Majumdar b,n a b
Birla Institute of Technology and Science-Pilani, Goa Campus, Zuarinagar, Goa 403726, India S N Bose National Centre for Basic Sciences, Sector III, Block JD, Kolkata 700 098, West Bengal, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 September 2010 Received in revised form 21 October 2010 Available online 28 October 2010
We present field and temperature dependence data on giant magnetoresistive (GMR) ion-beam sputtered Fe–Cr multilayers of varying Cr thickness. We show that the decrease in GMR with temperature is related to the decrease in sublattice magnetization due to thermal excitation of magnons in the antiferromagnetic configuration. The intralayer and the interlayer exchange energies thus obtained vary systematically as the Cr thickness increases. The corresponding decrease in the measured saturation field further supports our interpretation leading to a better understanding of the physics of GMR. & 2010 Elsevier B.V. All rights reserved.
Keywords: GMR Fe-Cr multilayers Exchange energies
1. Introduction The research on magnetism till late 20th century focused on its basic understanding and also on applications as soft and hard magnetic materials, both in crystalline and amorphous forms. In 1980s it was observed that surface and interface properties deviate considerably from those of the bulk. Then came, for the first time, the industrial application of electronic properties which depend on the spin of the electrons. In 1986 exchange coupling between ferromagnetic thin films across a non-magnetic metallic interlayer was realized in Fe–Cr and Co–Cu structures and rare earth-based multilayers giving rise to the so-called Giant Magnetoresistance (GMR¼(((r(T,H)–r(T,0))/r(T,0)) 100%)). Here the electrical resistivity decreased by almost 50% on the application of a magnetic field of a few kOe and the decrease was attributed to the spindependent bulk and interface scattering. The exchange coupling ¨ between Fe layers was established by Grunberg et al. [1] through scattering of light by spin waves. Visscher and Zhang [2] had shown that in the ferromagnetic configuration, realized by applying a magnetic field, the electrical resistivity decreases (negative GMR) from its value at zero or low fields when the Fe layers are antiferromagnetically aligned for appropriate Cr thickness. They computed the bulk scattering using the 2-current conduction model of Fert and Campbell [3] and found that the majority band of Fe has a much larger resistivity than the minority band. Butler et al. [4] used density functional theory for calculating the electronic structure and conductivity in many GMR systems. They
n
Corresponding author. E-mail address: [email protected] (A.K. Majumdar).
0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.10.036
found for Fe–Cr that the majority spin electrons at the Fermi level would see large difference between the atomic potentials of Fe and Cr and experience strong reflections from interfaces as well as strong impurity scattering at interdiffused Fe–Cr zones. Their calculations also showed that the density of states (DOS) at the mismatched spin channel was large. So the resistivity was much larger for the majority band due to both potential mismatch and higher DOS. A recent book [5] gives an excellent review on the physics and applications of GMR. Giant magnetoresistance (GMR) depends on various parameters of the multilayer structure, e.g. spacer layer thickness, interlayer surface roughness, etc. Petroff et al. [6] found that the GMR oscillated with the Cr thickness but varied rather slowly with the thickness of Fe. The ratio of the Cr thickness to the electron mean free path played a major role in determining the GMR. The scattering by Cr impurities in Fe layer was found to be less effective than the scattering by interface roughness. The scattering by Cr impurities was certainly spin dependent but probably less than the interface scattering. Varying interface roughness could considerably change GMR. There are many factors that might affect the temperature dependence of GMR. Thermal demagnetization of sublattice magnetization and spin-flip scattering by collision with spin waves are the few important factors. Hasegawa [7] studied the GMR in magnetic multilayers at finite temperatures with the use of the functional-integral method in the static approximation. The effect of spin fluctuation was shown to play an important role in the temperature dependence of the GMR. The temperature dependence of the calculated GMR was much stronger than that of the magnetization and it showed an almost linear decrease near the Curie temperature when the temperature was raised.
R.S. Patel, A.K. Majumdar / Journal of Magnetism and Magnetic Materials 323 (2011) 646–649
647
In an earlier work [8], we had investigated, among many other things, the temperature dependence of resistivity and magnetoresistance of three samples of Xe-ion-beam sputtered 30 bi-layers of ˚ ˚ Most importantly, the nearly fixed thickness Fe(20 A)/Cr( 12 A). sublattice magnetization m(T), a measure of the aniferromagnetic (AF) order parameter in the multilayer, was calculated using a simplified Hubbard model in terms of the planar and interlayer exchange energies. We found that the additional spin-dependent resistivity in the AF configuration (H¼0), proportional to m(T), decreased with increasing temperature in a rather complicated fashion but much the same way as our experimental data of the ˚ ˚ sample. Fe(20 A)/Cr( 12 A) The motivation behind the present work is to understand the inplane and out-of-plane exchanges, in Fe–Cr multilayers of variable ˚ keeping Fe fixed at 20 A˚ through the temCr thickness (6–14 A) perature dependence of the GMR. We find that the intralayer and the interlayer exchange energies, obtained by fitting the temperature dependence of the GMR to our earlier theory [8], are consistent with the Cr thickness variation of the measured Hsat, the field at which the GMR (H) saturates.
2. Experimental Details ˚ ˚ 30 bi-layers of Fe–Cr of composition Si/Cr(50 A)/[Fe(20 A)/ ˚ 30/Cr(50-t (A)), ˚ Cr(t A)] where t ¼6, 8, 10, 12 and 14 A˚ were grown on Si substrates by ion-beam sputter deposition technique [9] using 1100 eV Xe ions of 30 mA beam current. Electrical resistance was measured using the standard 4-probe dc-method with a resolution of 3 parts in 106. However, the resistivity values were accurate to only within 5–10% due to uncertainties in the measurements of the sample dimensions. The typical dimensions of the samples for the measurements were 3 8 mm2 and the current used was 10 mA. Magnetic field, provided by a Varian 15 in. (Model V-3800) electromagnet was always applied in the plane of the sample parallel to the current. The in-plane magnetoresistance (MR) was measured at every 100 mK with a stability of 10 mK from 2.3 to 300 K up to H¼16 kOe.
3. Results and discussion Fig. 1 shows magnetoresistance (MR) versus H plots at 4.2 and ˚ The MR saturates around 11 kOe. 300 K for sample 2 (Cr ¼8 A). Similar plots give Hsat ¼11, 11, 8 and 5.5 kOe for samples 2–5 with ˚ respectively. The samples on the average Cr ¼8, 10, 12 and 14 A, have residual resistivity of 30 mO cm, Residual Resistivity Ratio (RRR¼R(300 K)/R (4.2 K)) of 1.8 at Hsat and GMR of 32% at 4.2 K. Small angle X-ray reflectivity measurements [10] quantitatively ˚ show that the roughness of both Fe and Cr layers is (5.570.5) A. The grazing incidence X-ray diffraction (GIXD) measurements [10] on these samples have been performed by keeping the angle of incidence fixed at around yi E11 and making the detector scan over a large angular region. All the samples, in the limited angular region from 40 to 501, exhibit broad diffraction maxima at around 2y E44.51. No crystalline peaks for Fe or Cr are observed over the entire scan. The existence of broad peaks signifies that all the multilayer samples are polycrystalline in nature and the estimated grain size is found to be of the order of 15 nm. Before presenting our data on the temperature dependence of GMR, let us summarize the theory we are going to use in its interpretation. The GMR in multilayers is mainly due to the different magnetic configurations in H¼0 and H¼Hsat state. The temperature dependence of the GMR depends upon the number of magnons in the two states. We can very well assume that the temperature dependence of the phonon contribution is the same in
˚ at 4.2 Fig. 1. Magnetoresistance (MR) versus magnetic field H for sample 2 (Cr¼8 A) and 300 K. Black and red points are the data points in magnetic field increment and decrement cycles showing some low-field hysteresis. HSat 11 kOe for this sample. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
these two states. Thermal excitation of magnons, which causes local spin disorder in multilayers, will be different in the two configurations. We observe that the net magnetization changes very little (1% till 100 K and 5% till 300 K) in the presence of Hsat, i.e., the sublattice magnetization in the FM configuration is almost constant with temperature. For thermal excitation of spin waves in highly anisotropic antiferromagnets with weak interlayer antiferromagnetic coupling between the planes, Singh et al. [11] found that at low temperatures /nSTpT2. The extension of this theory in the present case, where each Fe layer is ferromagnetic and is antiferromagnetically coupled to the neighboring Fe layer, in terms of the planer (intralayer) and interlayer exchange energies, Jp and Jz, respectively, gives [8]
DrðTÞ 1 T ¼ 1 2 Drð0Þ p Jp
Z p=2 0
dqz ln
1 1eJz =T ð1cos2 qz Þ1=2
! ð1Þ
where qz is an interlayer magnon vector. This equation shows that the MR decreases approximately as T2 at low temperatures (T5Jz) and there is a cross-over to an approximately linear (T ln T) behavior at higher temperatures (T bJz). The function 1 (T2 or T ln T) has always a convex upward curvature. In Fig. 2 we have plotted Dr ¼ r(T, H¼0) r(T, H¼Hsat) versus T data for sample 3. The temperature ranges in the analysis are till 150, 200, 250 and 300 K. The solid lines are the least-squares fitted curves for fits to Eq. (1). It can be seen that the fit is excellent for the range To150 K. The fit becomes worse for higher temperature ranges. It is also clearly seen that Dr(T) has concave upward curvature near 300 K. The reason could be as follows: in the derivation of Dr(T) it is assumed that the sublattice magnetization in the FM configuration of the multilayers is almost constant with temperature but the change in magnetization in the AF configuration is huge and this is the governing factor for Dr(T). But in reality there is some change in sublattice magnetization with temperature in the FM configuration especially at high temperatures. Magnetization changes by 1% till 100 K but by 5% till 300 K. This change in magnetization in the FM configuration leads to the observed deviation from the predicted functional dependence of Dr(T), as given by Eq. (1). For this reason we restricted our analysis up to only 150 K for finding out some meaningful values of the parameters, Jp and Jz. The integral in Eq.(1) was evaluated numerically and used in a
648
R.S. Patel, A.K. Majumdar / Journal of Magnetism and Magnetic Materials 323 (2011) 646–649
Fig. 2. Dr ¼ r(T, H¼ 0)–r(T, H¼ Hsat) versus temperature (T) data for sample 3. The temperature ranges in the analysis are till 150, 200, 250 and 300 K. The solid lines are the least-squares fitted curves for fits to Eq. (1).
Table 1 Values of w2, correlation coefficient R2, the parameters Jp and Jz from fits to Eq. (1) till 150 K for samples 2, 3, 4 and 5.
Fig. 3. Dr ¼ r(T, H ¼0)–r(T, H¼Hsat) versus temperature (T) data for samples 2, 3, 4 and 5. The solid lines are the least-squares fitted curves for fits to Eq. (1).
3-parameter least squares fit program. Fig. 3 shows the best-fit curves of Dr(T) data to Eq. (1) till 150 K for samples 2, 3, 4 and 5. The values of w2 are consistent with the experimental resolution and the correlation coefficient R2 40.99999. The values of w2, R2 and the parameters, Jp and Jz are presented in Table 1. The parameter Jp that represents the planar/ intralayer exchange energy is much below the Curie temperature (1040 K) of bulk Fe. However, it may be closer to the unknown Curie temperature of 20 A˚ thick Fe film. The parameter Jz, the interlayer exchange energy, is found to be rather close to the spin-glass transition temperatures derived from the low-field magnetization and acsusceptibility measurements [12]. From Table 1 we find that Jz decreases gradually from 117 to 45 K and Jp increases from 201 to 330 K as Cr thickness increases from 8 A˚ (sample 2) to 14 A˚ (sample 5).
Samples
w2 (10 6)
R2
Jp (K)
Jz (K)
2 3 4 5
4.45 5.43 3.55 12.90
0.999996 0.999995 0.999996 0.999987
2017 7 213 7 8 265 7 6 3307 16
117 76 102 76 76 75 45 75
This is because of the decrease in the antiferromagnetic coupling between the Fe layers as seen from the corresponding decrease of the measured Hsat from 11 to 5.5 kOe. In an earlier work on the temperature dependence of the GMR Suzuki and Taga [13] found that Dr(T)¼ r(H¼0, T)–r(H¼Hsat, T) varies as T3/2 in the temperature range 2rT r300 K for Co/Cu superlattice. Their interpretation was based on the spin-mixing concept, viz., with increasing temperature spin-up and spin-down channels could not be assumed to be independent and there would be spin mixing. Dr(T) depended on the difference in population between the majority and minority spin electrons at T. With increasing temperature, this population difference decreased and so did Dr(T). Mattson et al. [14] found in magnetron sputtered Fe–Cr superlattice that the magnetoresistivity of the antiferromagnetically coupled films fell from its maximum value at low temperatures with a T2 dependence, while a ferromagnetically (FM) coupled film showed a T3/2 behavior. These power laws were explained as a consequence of the thermal excitation of magnons. Camley and Barnas [15] had suggested that the temperature dependence of the GMR could be attributed to the variation of the mean free path (MFP) with temperature.
4. Conclusions To conclude, the temperature dependence of the GMR is satisfactorily explained in terms of the thermal excitation of spin
R.S. Patel, A.K. Majumdar / Journal of Magnetism and Magnetic Materials 323 (2011) 646–649
waves in the ferromagnetic Fe layers with weak interlayer antiferromagnetic coupling between the neighboring Fe layers. We also show that the planar/intralayer exchange energy, Jp increases with increasing Cr thickness while the interlayer exchange energy, Jz decreases with it. The corresponding decrease in the measured Hsat, the saturation field, further supports this interpretation and strengthens the physics behind GMR.
Acknowledgements We acknowledge D. Temple and C. Pace of MCNC of North Carolina, USA, for providing the samples and the Department of Science and Technology, Government of India, for financial assistance (Project no. SR/S2/CMP-18/204). References ¨ [1] P. Grunberg, R. Schreiber, Y. Pang, M.B. Brodsky, H. Sowers, Phys. Rev. Lett. 57 (1986) 2442.
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