Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 1(11), 56-58, November (2012) Res.J.Recent Sci. International Science Congress Association 56 Short Communication Drude Formalism Study of The Giant Magnetoresistance in Fe(t)/Cu/Fe TrilayersUba J.I.1*, Ekpunobi A.J.2 and Ekwo P.I.1*Department of Physics, Nwafor Orizu College of Education, P.M.B. 1734, Onitsha, Anambra, NIGERIA Department of Physics/Industrial Physics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra, NIGERIAAvailable online at: www.isca.in Received 6th September 2012, revised 18th October 2012, accepted 20th October 2012Abstract The recently developed Drude-like model for resistivity in metallic multilayers is applied to Fe(t)/Cu(10A)/Fe(10A) trilayer systems. The basic GMR characteristics deduced conform qualitatively to reports communicated in literature. Our analyses show that the Fuchs-Sondheimer theory of thin film resistivity is not valid for layered structures and spin-dependent scattering in the bulk is the dominant mechanism for GMR in the investigated systems. Keywords: GMR, relaxation time, trilayer. IntroductionTwo decades after its discovery in Fe-Cr trilayers1,2, the giant magnetoresistance still attracts growing number of theoretical and experimental studies. Several of these investigations seek better GMR materials3–5 while more in depth understanding of the spin-dependent transport is the focus of most theoretical studies6–7. Some of the influencing factors like pressure and temperature have been investigated as well8–11. The theoretical models developed over the years to interpret GMR, consists of two main approaches: quasiclassical12,13 and quantum methods14,15, both evaluate resistivity on per-layer basis. However Camblong16 in his Kubo formalism approach demonstrated the equivalence of both methods. A generalized proof of this has been developed recently17. With regards to metallic structures, it is well known that in the homogenous limit  ( is mean free path, = sublayer index, = , spin and = thickness of th sublayer ) an electron probes all scattering within a mean free path which includes several layers and thus averages all sorts of scattering medium16. It is then conceivable to have a model that yields the global resistivity of a structure on a non per-layer basis. In this respect, a Drude-like model was recently developed17 in the framework of piece-wise constant potentials. The present contribution centers on the application of the model to the GMR of Fe(t)/Cu(10A)/Fe(10A) trilayer systems. Section II presents a brief description of the model and in section III we present results of the numerical analyses. Material and Methods Resistivity of either spin channel is given by (1) where the parameters have the same meanings as in the conventional Drude model but now with respect to spin . Also the relaxation time is dependent on the magnetic state (antiparallel and parallel alignments) of the system. The spin channel density and fermi velocity are respectively given by   (2) And,  "#  (3) Where denotes total spin-dependent potential of the system and consists of the bulk potential, potential step at interfaces and s-d scattering potential introduced by interface roughness. E is the adjustable fermi level. Equation 1 is solved for both magnetic states and the global resistivity evaluated using the two current model. Results and DiscussionFrom what has been shown, resistivity is dependent on the magnetic states through relaxation time. To characterize the basic features, we adopt bulk potentials of the constituent layers as given in12 and assume effective mass M= M= 4× free electron mass. We analyse numerically, five trilayers of Fe(t)/Cu (10Å) /Fe(10Å) where t = 5 – 25Å in steps of 5Å. Our results compare positively with what has been reported about GMR in experiments and theoretical models. The MR shows decreasing trend with increasing structure thickness as illustrated in figure 1. This is due to increasing tFe; a well known fact15 that is explicitly shown in figure 2a. Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 1(11), 56-58, November (2012) Res. J. Recent Sci. International Science Congress Association 57 Figure-1 GMR as function of thickness of structure (a) (b) Figure-2 (a) GMR as function of Fe thickness for tFe = 5, 10, 15, 20, 25Å. (b) GMR as function of mean free path calculated for = (+ )/2 in parallel alignment As expected, the MR maximizes in the limit L,tFe15 as demonstrated in figure 2b, where is arithmetic mean of the mean free paths of the two spin channels in the trilayers. It is observed from figure 1 – 2b that MRmax of 0.24 occurred at L = 25Å, tFe = 5Å and = 97Å. In comparison, figure 2b is qualitatively analogous to those of Fe-Cr structures of unit specularity factor at outer boundaries13. This inherent unit specularity is a natural consequence of confining conduction electrons within piece-wise potential environment as adopted in our model. The Fuchs-Sondheimer theory18,19 had predicted resistivities of thin and thick films as decreasing functions of thickness due to decreasing surface effect. However resistivities shown in figure 3 contradict this theory as is the case in reference 15 and 18. Figure-3 Antiparallel and parallel alignment resistivities as function of structure thickness One can then infer that dominant contribution to GMR in the present case comes from bulk effects. Conclusion We have successfully applied a drude-like model to Fe/Cu/Fe trilayer system. The qualitative agreement of the derived basic characteristics of GMR with those already communicated in many experimental and theoretical reports give credibility to the model. The model emphasizes the role of relaxation time and has the merit of straight forward evaluation of resistivities on a non-per-layer basis and as such is computationally less demanding unlike other models. Reference1.Baibich M.N., Broto J.M., Fert A., Nguyen Van Dau F., Petroff F., Eitenne P., Creuzet G., Friederich A. and Chazelas J., Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices, Phys. Rev. Lett61, 2472 - 2475 (1988) Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 1(11), 56-58, November (2012) Res. J. Recent Sci. International Science Congress Association 58 2.Binasch G., Grunberg P., Saurenbach F. and Zinn W., Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange, Phys. Rev. B, 39, 4828 - 4830 (1989)3.Lu L., Lu G., Zhang Z., Gao C., Yu T. and Chen P., Giant magnetoresistance of Co/ITO multilayers, Solid State Commun.,149, 2254-2256 (2009)4.Papp G. and Borza S., Giant magnetoresistance in a two-dimensional electron gas modulated by periodically repeated magnetic barriers, Solid State Commun.,150, 2023 - 2027 (2010)5.Lu M-W. and Yang G-J., Magnetoresistance effect in a hybrid ferromagnetic/semiconductor nanostructure, Solid State Commun.,141, 248 - 251 (2007)6.Elsafi B., Trigui F. and Fakhfakh Z., Effects of bulk and interface scattering on giant magnetoresistance in the Co/Cu multilayer systems, Comput. Mater. Sci., 50(2), 800 - 804 (2010)7.Vedyayev A., Ryzhanova N. and Dieny B., Quantum effects in the giant magnetoresistance (GMR) of magnetic multilayers, Physica A, 241, 207 - 215 (1997)8.Oomi G., Sakai T., Uwatoko Y., Takanashi K. and Fujimori H., Magnetoresistance of magnetic multilayers at high pressure, Physica B,239, 19 - 28 (1997)9.Yu. T, Li X-Q., Li D-G., Hao S-F., Wang L-M., Zhang Z-G., Wu G.H., Zhang X.X., Li Q-L. and Chen P., Magnetic property and magnetoresistance in Fe/ITO multilayers, J. Magn. Magn. Mater., 320, 2185 - 2189 (2008)10.Lu L., Yang Y-X., Gao C., Xiong Y-Q. and Chen P., Temperature dependence of magnetoresistance in Co/ITO multilayers, J. Alloys. Comp., 492, 61 - 64 (2010)11.Uba I., Ekpunobi A.J. and Ekwo P.I., Magnetoresistance – temperature relationship: calculus of variation approach, J. Sci. and Arts,4(17), 509 - 512 (2011)12.Hood R.Q. and Falicov L.M., Boltzmann equation approach to the negative magnetoresistance of ferromagnetic – normal metal multilayers, Phys. Rev. B,46, 8287 - 8296 (1992)13.Barnas J., Fuss A., Camley R.E., Grunberg P. and Zinn W., Novel magnetoresistance effect in layered magnetic structures: Theory and experiment, Phys. Rev. B,42, 8110 - 8120 (1990)14.Camblong H.E. and Levy P.M., Novel results for quasiclassical linear transport in metallic multilayers, Phys. Rev. Lett., 69, 2835 - 2838 (1992)15.Barnas J and Bruynseraede Y., Electronic transport in ultrathin magnetic multilayers, Phys. Rev. B53, 5449 - 5460 (1996)16.Camblong H.E., Linear transport theory of magnetoconductance in metallic multilayers: A real – space approach, Phys. Rev. B,51, 1855 - 1865 (1995)17.The model and its derivation are part of still – in – progress doctorate work of U.J.I. 18.Thanh N.T., Tu L.T., Ha N.D., Kim C.O., Kim C., Shin K.H. and Parvatheeswara Rao B., Thickness dependence of parallel and perpendicular anisotropic resistivity in Ta/NiFe/IrMn/Ta multilayer studied by anisotropic magnetoresistance and planar Hall effect, J. Appl. Phys.,101, 053702-1-5 (2007)19.Sondheimer E.H., The mean free path of electrons in metal, Adv. Phys.50(6), 466 - 537 (2001)