The magnetoresistance tensor of La(0.8)Sr(0.2)MnO(3)
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t The magnetoresistance tensor of La . Sr . MnO Y. Bason , ∗ J. Hoffman , C. H. Ahn , and L. Klein Department of Physics, Nano-magnetism Research Center,Institute of Nanotechnology and Advanced Materials,Bar-Ilan University, Ramat-Gan 52900, Israel and Department of Applied Physics, Yale University, New Haven, Connecticut 06520-8284, USA (Dated: November 3, 2018)We measure the temperature dependence of the anisotropic magnetoresistance (AMR) and theplanar Hall effect (PHE) in c-axis oriented epitaxial thin films of La . Sr . MnO , for differentcurrent directions relative to the crystal axes, and show that both AMR and PHE depend stronglyon current orientation. We determine a magnetoresistance tensor, extracted to 4 th order, whichreflects the crystal symmetry and provides a comprehensive description of the data. We extend theapplicability of the extracted tensor by determining the bi-axial magnetocrystalline anisotropy inour samples. PACS numbers: 75.47.-m, 75.47.Lx, 72.15.Gd, 75.70.Ak
The interplay between spin polarized current and mag-netic moments gives rise to intriguing phenomena whichhave led to the emergence of the field of spintronics [1].In most cases, the materials used for studying these phe-nomena have been amorphous alloys of 3d itinerant fer-romagnets (e.g., permalloy), while much less is knownabout the behavior in materials which are crystalline andmore complicated. Manganites, which are magnetic per-ovskites, serve as a good example for such a system. Aswe will show, elucidating these phenomena in this ma-terial system provides tools for better theoretical under-standing of spintronics phenomena and reveals opportu-nities for novel device applications.The magnetotransport properties of manganitesknown for their colossal magnetoresistance have beenstudied quite extensively; nevertheless, along numerousstudies devoted to elucidating the role of the magni-tude of the magnetization, relatively few reports have ad-dressed the role of the orientation of the magnetization,which is known to affect both the longitudinal resistivity ρ long (anisotropic magnetoresistance effect - AMR) andtransverse resistivity ρ trans (planar Hall effect - PHE).For conductors that are amorphous magnetic films, thedependence of ρ long and ρ trans on the magnetic orienta-tion is given by: ρ long = ρ ⊥ + ( ρ k − ρ ⊥ ) cos ϕ (1)and ρ trans = ( ρ k − ρ ⊥ ) sin ϕ cos ϕ (2)where ϕ is the angle between the current J and the mag-netization M and ρ k and ρ ⊥ are the resistivities paralleland perpendicular to M , respectively [3, 4]. Eqs. 1 and2 are not expected to apply to crystalline conductors,as they are independent of the crystal axes [5]. Never-theless, they have been used to describe AMR and PHE in epitaxial films [6, 7, 8, 9]; qualitative and quantita-tive deviations were occasionally attributed to extrinsiceffects.Here, we quantitatively identify the crystalline con-tributions to AMR and PHE in epitaxial films ofLa . Sr . MnO (LSMO) and replace Eqs. 1 and 2 withequations that provide a comprehensive description ofthe magnetotransport properties of LSMO. The equa-tions are derived by expanding the resistivity tensor to4 th order and keeping terms consistent with the crystalsymmetry.AMR and PHE in manganites constitute an impor-tant aspect of their magnetotransport properties; hence,quantitative determination of these effects is essential forcomprehensive understanding of the interplay betweenmagnetism and transport in this class of materials. Inaddition, when the dependence of AMR and PHE on lo-cal magnetic configurations is known, the two effects canbe used as a powerful tool for probing and tracking staticand dynamic magnetic configurations in patterned struc-tures. Moreover, as the magnitude of the AMR and PHEchanges dramatically with current direction, the elucida-tion of the appropriate equations is crucial for designingnovel devices with optimal properties that are based onthese phenomena.Our samples are epitaxial thin films ( ∼
40 nm) ofLSMO with a Curie temperature (T c ) of ∼
290 K grownon cubic single crystal [001] SrTiO substrates using off-axis magnetron sputtering. θ − θ x-ray diffraction revealsc-axis oriented growth (in the pseudocubic frame), withan out-of-plane lattice constant of ∼ . ∼ . . o . The film surfaces have been characterized us-ing atomic force microscopy, which shows a typical root-mean-square surface roughness of ∼ . θ relative to the [100] direction( θ = 0 ◦ , ◦ , ◦ , ◦ , ◦ , ◦ , ◦ ), with electrical leadsthat allow for AMR and PHE measurements.Fig. 1 presents ρ long and ρ trans data obtained by ap-plying a field of H=4 T in the film plane and rotatingthe sample around the [001] axis. The figure shows thedata for all seven patterns at T=5, 125 and 300 K. AtT = 300 K both ρ long and ρ trans seem to behave accord-ing to Eqs. 1 and 2. However, contrary to these equa-tions, the amplitude of ρ long differs from the amplitudeof ρ trans ; moreover, they both change with θ , the anglebetween J and [100].The discrepancies increase as the temperature de-creases, and at T=125 K the variations in the amplitudesfor measurements taken for different θ increase. Further-more, the location of the extremal points are dominatedby α , the angle between M and [100]. At T = 5 K, thedeviations are even more evident as the AMR measure-ments are no longer described with a sinusoidal curve. Allthese observations clearly indicate the need for a higherorder tensor to adequately describe the magnetotrans-port behavior of LSMO.The resistivity tensor in a magnetic conductor dependson the direction cosines, α i , of the magnetization vector,and can be expressed as a series expansion of powers ofthe α i [10], giving: ρ ij ( α ) = X k,l,m... =1 ( a ij + a kij α k + a klij α k α l ++ a klmij α k α l α m + a klmnij α k α l α m α n + ... ) (3)where i, j = 1 , , a ’s are expansion coefficients.As usual ρ ij ( α ) = ρ sij ( α ) + ρ aij ( α ) where ρ sij and ρ aij aresymmetric and antisymmetric tensors, respectively. Asboth AMR and PHE are symmetric, we use only ρ sij fortheir expression. As we are interested only in the in-plane properties, we use the tensor expansion for crystalswith m3m cubic-crystal structure [11]. The 4 th ordersymmetric resistivity tensor ρ s for this class of materialsin the xy plane (as M , J and the measurements are allin the plane of the film) is given by: ρ s = (cid:18) C ′ + C ′ α + C ′ α C ′ α α C ′ α α C ′ + C ′ α + C ′ α (cid:19) . (4)When J is along θ we obtain: ρ long = A cos(2 α − θ ) + B cos(2 α + 2 θ ) + C cos(4 α ) + D (5)and ρ trans = A sin(2 α − θ ) − B sin(2 α + 2 θ ) (6)with: A = ( C ′ + C ′ + C ′ ) / B = ( C ′ + C ′ − C ′ ) / C = C ′ / D = C ′ + C ′ / C ′ / A , B , C and D ) with which we fit (as shown in Figure 1) at anygiven temperature and magnetic field a set of 14 differentcurves (7 AMR curves and 7 PHE curves).The parameter A is a coefficient of a term describinga non-crystalline contribution since ( α − θ ) is the anglebetween M and J irrespective of their orientation relativeto the crystal axes. On the other hand, the parameters B and C are coefficients of terms that depend on theorientation of M and/or J relative to the crystal axes.We note that adding the terms with the coefficient B (in both Eq. 5 and 6) to the ” A ” term changes only theamplitude and the phase of the signal compared to Eqs.1 and 2: Eq. 5 can be written (for C=0) as: ρ long = E cos(2 α − φ long ) + D (7)where E = A + B + 2 AB cos 4 θ and sin φ long = A − BE sin(2 θ ); and Eq. 6 can be written as: ρ trans = F sin(2 α − φ trans ) (8)where F = A + B − AB cos 4 θ and sin φ trans = A + BF sin 2 θ . The amplitude of ρ trans ( α ), F , varies with θ between a maximal value of | A + B | for θ = ± ◦ and aminimal value of | A − B | for θ = 0 , ± ◦ . On the otherhand, the amplitude of ρ long ( α ), E , obtains its maximalvalue | A + B | at θ = 0 , ± ◦ and its minimal value | A − B | at θ = ± ◦ . When the C term is added it does not affect ρ trans ; however, ρ long behaves qualitatively differently.We thus observe that the current direction affects quitedramatically the amplitude of the effect. At 125 K, forinstance, the PHE amplitude for current at 45 ◦ relativeto [100] is more than 20 times larger than the PHE forcurrent parallel to [100]. This means that appropriateselection of the current direction that takes into consid-eration crystalline effects is important for designing de-vices that use the PHE for magnetic sensor or magneticmemory applications [12].Figure 2 presents the temperature dependence of B/A and
C/A . Close to T c both B and C are negligible rel-ative to A ; therefore, AMR and PHE measurements ap-pear to fit Eqs. 1 and 2. At intermediate temperatureswhere C is still much smaller than A (while B and A are of the same order), the signal remains sinusoidal, al-though its deviation from Eqs. 1 and 2 becomes quiteevident. At low temperatures, C is on the order of B ,and the AMR signal is no longer sinusoidal.When AMR and PHE measurements are performedwith low applied fields, M is no longer parallel to H , dueto intrinsic magnetocrystalline anisotropy. Our LSMOfilms exhibit bi-axial magnetocrystalline anisotropy witheasy axes along h i directions, a manifestation of in-plane cubic symmetry. When a field H is applied,the total free energy consists of the magnetocrystallineanisotropy energy and the Zeeman energy: E = K α − M H cos( α − β ) (9)where K is the magnetocrystalline anisotropy energyand β is the angle between H and [100]. The first term isresponsible for the bi-axial magnetocrystalline anisotropywith easy axes along α = ± π and α = ± π . We havedetermined the value of K at various temperatures (seeFig. 2) by switching the magnetization between the twoeasy axes (see Fig. 3). The extracted value of K allowsus by using Eqs. 5, 6 and 9 to fit the AMR and PHEdata obtained with relatively low applied fields (e.g., 500Oe), where M does not follow H (see Fig. 3).In summary, we have expanded the magnetoresistancetensor to 4 th order keeping terms consistent with thesymmetry of epitaxial films of LSMO and derived equa-tions that provide a comprehensive description of AMRand PHE in LSMO films in a wide range of tempera-tures. The results shed new light on the interplay be-tween magnetism and electrical transport in this class ofmaterials and may serve as a basis for further study ofthe microscopic origin of magnetotransport properties ofLSMO and other manganites. The results contribute tothe ability to monitor magnetic configurations via mag-netotransport properties, a feature of particular impor-tance in studying nano-structures, and will facilitate thedesign of novel devices that use AMR and PHE. We acknowledge useful discussions with E. Kogan.L.K. acknowledges support by the Israel Science Founda-tion founded by the Israel Academy of Sciences and Hu-manities. Work at Yale supported by NSF MRSEC DMR0520495, DMR 0705799, NRI, ONR, and the PackardFoundation. ∗ Electronic address: [email protected][1] G. A. Prinz, Science , 1660 (1998).[2] M. McCormack, S. Jin, T H. Tiefel, R. M. Fleming, J.M. Phillips and R. Ramesh, Appl. Phys. Lett. , 3045(1994);[3] C. Goldberg and R. E. Davis, Phys. Rev. , 1121 (1954);F. G. West, J. Appl. Phys. , 1171 (1963); W. M. Bullis,Phys. Rev. , 292 (1958).[4] T. R. McGuire and R. I. Potter, IEEE Trans. Magn. ,1018 (1975).[5] D¨oring, W., Ann. Physik (
32) (1938) 259.[6] X. Jin, R. Ramos, Y. Zhou, C. McEvoy, and I. V. Shvets,J. Appl. Phys. , 08C509 (2006);[7] H. X. Tang, R. K. Kawakami, D. D. Awschalom, and M.L. Roukes, PRL ,107201 (2003);[8] Y. Bason, L. Klein, J.-B. Yau, X. Hong, and C. H. Ahn,Appl. Phys. Lett. , 2593 (2004).[9] I.C. Infante, V. Laukhin, F. S´anchez, J. Fontcuberta, Ma-terials Science and Engineering B (2006) 283-286;[10] T. T. Chen, V. A. Marsocii, Physica (1972) 498-509.[11] Birss, R. R., Symmetry and Magnetizm, North-HollandPubl. Comp. (Amsterdam, 1964).[12] Y. Bason, L. Klein, J.-B. Yau, X. Hong, J. Hoffman, andC. H. Ahn, J. Appl. Phys. , 08R701 (2006). α [Deg.]354.2354.6355 T=5 K882883884885 ρ l ong [ µ Ω c m ] T=125 K -0.200.2
T=5 K -20020 0 45 90 135 180 θ =0 o θ =15 o θ =30 o θ =45 o θ =60 o θ =75 o θ =90 o α [Deg.] T=300 K -101 ρ t r an s [ µ Ω c m ] T=125 K
FIG. 1: Longitudinal resistivity ρ long (left) and transverse resistivity ρ trans (right) vs. α , the angle between the magnetizationand [100], for different angles θ (the angle between the current direction and [100]) at different temperatures with an appliedmagnetic field of 4 T. The solid lines are fits to Eqs. 5 and 6. Inset: Sketch of the relative orientations of the current density J , magnetization M , and the crystallographic axes. -0.200.20.40.60.811.2 -500501001502002500 50 100 150 200 250 300B/AC/A K K [ kJ / m ] T [K] C oe ff i c i en t s r a t i o FIG. 2: The ratios of the coefficients from Eqs. 5 and 6 (B/A and C/A) (left axis) and the coefficient K from Eq. 9 (rightaxis) as a function of temperature. -0.15-0.1-0.0500.050.10.15 0 90 180 270 360 ρ t r an s [ µ Ω c m ] β [Deg.] -0.014-0.012-0.01-0.008-0.006-0.004-0.00200.002 ρ t r an s [ µ Ω c m ] Magnetic Field [Oe]